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0
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2
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1
ISI2016DCG5
If $\tan\: x=p+1$ and $\tan\; y=p1,$ then the value of $2\:\cot\:(xy)$ is $2p$ $p^{2}$ $(p+1)(p1)$ $\frac{2p}{p^{2}1}$
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isi2016dcg
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1
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2
ISI2015MMA16
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\: (\geq 2)$ and $n\: (\geq 1)$ respectively, satisfy $f(x^2+1)=f(x)g(x),$ for every $x \in \mathbb{R}$, then $f$ has exactly one real root $x_0$ such that $f’(x_0) \neq 0$ $f$ has exactly one real root $x_0$ such that $f’(x_0) = 0$ $f$ has $m$ distinct real roots $f$ has no real root
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isi2015mma
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1
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3
ISI2015MMA18
The set of complex numbers $z$ satisfying the equation $(3+7i)z+(102i)\overline{z}+100=0$ represents, in the complex plane, a straight line a pair of intersecting straight lines a point a pair of distinct parallel straight lines
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isi2015mma
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2
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4
ISI2015MMA5
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is $2^n$ $2^{n+1}$ $2^{n1}$ $2^{2n}$
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Oct 9
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1
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5
ISI2015MMA21
Let $\omega$ denote a complex fifth root of unity. Define $b_k =\sum_{j=0}^{4} j \omega^{kj},$ for $0 \leq k \leq 4$. Then $ \sum_{k=0}^{4} b_k \omega ^k$ is equal to $5$ $5 \omega$ $5(1+\omega)$ $0$
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Oct 9
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6
ISI2015MMA3
Let $x$ be a positive real number. Then $x^2+\pi ^2 + x^{2 \pi} > x \pi+ (\pi + x) x^{\pi}$ $x^{\pi}+\pi^x > x^{2 \pi} + \pi ^{2x}$ $\pi x +(\pi+x)x^{\pi} > x^2+\pi ^2 + x^{2 \pi}$ none of the above
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Oct 8
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7
ISI2016DCG36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$.. The number of bijective functions from $X$ to $Y$ is $n^{n}$ $n\log_{2}n$ $n^{2}$ $n!$
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8
ISI2015MMA93
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$
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Oct 5
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shafique
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6
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isi2015mma
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1
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9
ISI2015MMA94
Let $G$ be the group $\{\pm1, \pm i \}$ with multiplication of complex numbers as composition. Let $H$ be the quotient group $\mathbb{Z}/4 \mathbb{Z}$. Then the number of nontrivial group homomorphisms from $H$ to $G$ is $4$ $1$ $2$ $3$
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Oct 5
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shafique
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10
ISI2015MMA79
Let $g(x,y) = \text{max}\{12x, 8y\}$. Then the minimum value of $g(x,y)$ $ $ as $(x,y)$ varies over the line $x+y =10$ is $5$ $7$ $1$ $3$
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Oct 4
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1
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11
ISI2015MMA78
The value of $\underset{n \to \infty}{\lim} \bigg[ (n+1) \int_0^1 x^n \text{ln}(1+x) dx \bigg]$ is $0$ $\text{ln }2$ $\text{ln }3$ $\infty$
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12
ISI2015MMA28
In a triangle $ABC$, $AD$ is the median. If length of $AB$ is $7$, length of $AC$ is $15$ and length of $BC$ is $10$ then length of $AD$ equals $\sqrt{125}$ $69/5$ $\sqrt{112}$ $\sqrt{864}/5$
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1
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13
ISI2015MMA22
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1 \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
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Oct 4
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14
ISI2015MMA19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$
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Oct 4
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isi2015mma
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15
ISI2015MMA13
The number of real roots of the equation $2 \cos \bigg( \frac{x^2+x}{6} \bigg) = 2^x +2^{x} \text{ is }$ $0$ $1$ $2$ infinitely many
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Oct 4
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1
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16
ISI2015MMA12
Consider the polynomial $x^5+ax^4+bx^3+cx^2+dx+4$ where $a,b,c,d$ are real numbers. If $(1+2i)$ and $(32i)$ are two two roots of this polynomial then the value of $a$ is $524/65$ $524/65$ $1/65$ $1/65$
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Oct 4
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1
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17
Where to scan the barcode given in GateOverflow book
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Oct 3
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shaktisingh
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219
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1
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18
ISI2016DCG61
The value of $\sin^{6}\frac{\pi}{81}+\cos^{6}\frac{\pi}{81}1+3\sin^{2}\frac{\pi}{81}\:\cos^{2}\frac{\pi}{81}$ is $\tan^{6}\frac{\pi}{81}$ $0$ $1$ None of these
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Oct 2
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isi2016dcg
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19
ISI2016DCG64
If $\cos2\theta=\sqrt{2}(\cos\theta\sin\theta)$ then $\tan\theta$ equals $1$ $1$ or $1$ $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$ or $1$ None of these
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Oct 2
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isi2016dcg
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1
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20
ISI2018DCG15
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is $6$ $9$ $12$ $18$
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Oct 1
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isi2018dcg
0
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1
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21
ISI2016DCG12
The highest power of $3$ contained in $1000!$ is $198$ $891$ $498$ $292$
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Oct 1
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6
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isi2016dcg
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1
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22
ISI2015MMA25
The limit $\underset{x \to \infty}{\lim} \bigg( \frac{3x1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
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Oct 1
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isi2015mma
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1
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23
ISI2015MMA20
The limit $\underset{n \to \infty}{\lim} \bigg( 1 \frac{1}{n^2} \bigg) ^n$ equals $e^{1}$ $e^{1/2}$ $e^{2}$ $1$
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Oct 1
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isi2015mma
+2
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1
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24
ISI2015MMA10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^31}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above
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Oct 1
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8
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isi2015mma
0
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1
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25
ISI2018DCG9
Let $f(x)=1+x+\frac{x^2}{2}+\frac{x^3}{3}...+\frac{x^{2018}}{2018}.$ Then $f’(1)$ is equal to $0$ $2017$ $2018$ $2019$
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Sep 30
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isi2018dcg
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1
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26
ISI2018DCG7
You are given three sets $A,B,C$ in such a way that the set $B \cap C$ consists of $8$ elements, the set $A\cap B$ consists of $7$ elements, and the set $C\cap A$ consists of $7$ elements. The minimum number of elements in the set $A\cup B\cup C$ is $8$ $14$ $15$ $22$
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Sep 30
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isi2018dcg
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1
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27
ISI2018DCG10
Let $f’(x)=4x^33x^2+2x+k,$ $f(0)=1$ and $f(1)=4.$ Then $f(x)$ is equal to $4x^43x^3+2x^2+x+1$ $x^4x^3+x^2+2x+1$ $x^4x^3+x^2+2(x+1)$ none of these
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Sep 30
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6
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isi2018dcg
0
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1
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28
ISI2016DCG66
If $\sin(\sin^{1}\frac{2}{5}+\cos^{1}x)=1,$ then $x$ is $1$ $\frac{2}{5}$ $\frac{3}{5}$ None of these
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Sep 30
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isi2016dcg
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1
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29
ISI2016DCG49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}1}{x^{\frac{1}{4}}1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
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Sep 28
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isi2016dcg
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1
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30
ISI2016DCG50
The domain of the function $\ln(3x^{2}4x+5)$ is set of positive real numbers set of real numbers set of negative real numbers set of real numbers larger than $5$
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Sep 28
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isi2016dcg
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31
ISI2016DCG53
$\underset{x\rightarrow1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
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Sep 28
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isi2016dcg
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1
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32
ISI2016DCG39
The medians $AD$ and $BE$ of the triangle with vertices $A(0,b),B(0,0)$ and $C(a,0)$ are mutually perpendicular if $b=\sqrt{2}a$ $a=\pm\sqrt{2}b$ $b=\sqrt{2}a$ $b=a$
answered
Sep 27
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isi2016dcg
0
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1
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33
ISI2015MMA53
The number of cars $(X)$ arriving at a service station per day follows a Poisson distribution with mean $4$. The service station can provide service to a maximum of $4$ cars per day. Then the expected number of cars that do not get service per day equals $4$ $0$ $\Sigma_{i=0}^{\infty} i P(X=i+4)$ $\Sigma_{i=4}^{\infty} i P(X=i4)$
answered
Sep 27
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isi2015mma
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1
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34
ISI2016DCG40
The equations $x=a\cos\theta+b\sin\theta$ and $y=a\sin\theta+b\cos\theta,(0\leq\theta\leq2\pi$ and $a,b$ are arbitrary constants$)$ represent a circle a parabola an ellipse a hyperbola
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Sep 26
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35
ISI2015MMA1
Let $\{f_n(x)\}$ be a sequence of polynomials defined inductively as $ f_1(x)=(x2)^2$ $f_{n+1}(x) = (f_n(x)2)^2, \: \: \: n \geq 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the coefficient of $x$ in $f_n(x)$. Then $a_n=4, \: b_n=4^n$ $a_n=4, \: b_n=4n^2$ $a_n=4^{(n1)!}, \: b_n=4^n$ $a_n=4^{(n1)!}, \: b_n=4n^2$
answered
Sep 26
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0
votes
1
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36
ISI2015MMA15
The number of real solutions of the equations $(9/10)^x = 3+xx^2$ is $2$ $0$ $1$ none of the above
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Sep 26
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isi2015mma
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1
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37
ISI2015MMA14
Consider the following system of equivalences of integers, $x \equiv 2 \text{ mod } 15$ $x \equiv 4 \text{ mod } 21$ The number of solutions in $x$, where $1 \leq x \leq 315$, to the above system of equivalences is $0$ $1$ $2$ $3$
answered
Sep 26
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isi2015mma
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1
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38
ISI2015MMA11
The number of positive integers which are less than or equal to $1000$ and are divisible by none of $17$, $19$ and $23$ equals $854$ $153$ $160$ none of the above
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Sep 26
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7
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isi2015mma
0
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1
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39
ISI2015MMA17
Let $X=\frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$. Then, $X \lt1$ $X\gt3/2$ $1\lt X\lt 3/2$ none of the above holds
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Sep 26
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isi2015mma
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2
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40
ISI2015MMA4
Suppose in a competition $11$ matches are to be played, each having one of $3$ distinct outcomes as possibilities. The number of ways one can predict the outcomes of all $11$ matches such that exactly $6$ of the predictions turn out to be correct is $\begin{pmatrix}11 \\ 6 \end{pmatrix} \times 2^5$ $\begin{pmatrix}11 \\ 6 \end{pmatrix} $ $3^6$ none of the above
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Sep 25
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arvin
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20
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isi2015mma
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