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Recent questions tagged binomial-theorem

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32
Kenneth Rosen Edition 7 Exercise 6.4 Question 15 (Page No. 421)
Show that $\binom{n}{k} \leq 2^{n}$ for all positive integers $n$ and all integers $k$ with $0 \leq k \leq n.$
Show that $\binom{n}{k} \leq 2^{n}$ for all positive integers $n$ and all integers $k$ with $0 \leq k \leq n.$
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34
Kenneth Rosen Edition 7 Exercise 6.4 Question 13 (Page No. 421)
What is the row of Pascal’s triangle containing the binomial coefficients $\binom{9}{k} ,\: 0 \leq k \leq 9?$
What is the row of Pascal’s triangle containing the binomial coefficients $\binom{9}{k} ,\: 0 \leq k \leq 9?$
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36
Kenneth Rosen Edition 7 Exercise 6.4 Question 11 (Page No. 421)
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x^{2} − \frac{1}{x}\right)^{100},$ where $k$ is an integer.
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x^{2} − \frac{1}{x}\right)^{100},$ where $k$ is an integer.
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37
Kenneth Rosen Edition 7 Exercise 6.4 Question 10 (Page No. 421)
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x + \frac{1}{x}\right)^{100},$ where $k$ is an integer.
Give a formula for the coefficient of $x^{k}$ in the expansion of $\left(x + \frac{1}{x}\right)^{100},$ where $k$ is an integer.
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45
Kenneth Rosen Edition 7 Exercise 6.4 Question 2 (Page No. 421)
Find the expansion of $(x + y)^{5}$ using combinatorial reasoning, as in Example $1.$ using the binomial theorem.
Find the expansion of $(x + y)^{5}$using combinatorial reasoning, as in Example $1.$using the binomial theorem.
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46
Kenneth Rosen Edition 7 Exercise 6.4 Question 1 (Page No. 421)
Find the expansion of $(x + y)^{4}$ using combinatorial reasoning, as in Example $1.$ using the binomial theorem.
Find the expansion of $(x + y)^{4}$using combinatorial reasoning, as in Example $1.$ using the binomial theorem.
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48
ISI2014-DCG-18
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals $2^n+n2^{n-1}$ $2^n-n2^{n-1}$ $2^n$ none of these
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals$2^n+n2^{n-1}$$2^n-n2^{n-1}$$2^n$none of these
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51
ISI2015-DCG-18
The value of $(1.1)^{10}$ correct to $4$ decimal places is $2.4512$ $1.9547$ $2.5937$ $1.4512$
The value of $(1.1)^{10}$ correct to $4$ decimal places is$2.4512$$1.9547$$2.5937$$1.4512$
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52
ISI2015-DCG-21
The value of the term independent of $x$ in the expansion of $(1-x)^2(x+\frac{1}{x})^7$ is $-70$ $70$ $35$ None of these
The value of the term independent of $x$ in the expansion of $(1-x)^2(x+\frac{1}{x})^7$ is$-70$$70$$35$None of these
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53
ISI2016-DCG-21
The value of the term independent of $x$ in the expansion of $(1-x)^{2}(x+\frac{1}{x})^{7}$ is $-70$ $70$ $35$ None of these
The value of the term independent of $x$ in the expansion of $(1-x)^{2}(x+\frac{1}{x})^{7}$ is$-70$$70$$35$None of these
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54
ISI2017-DCG-11
The coefficient of $x^6y^3$ in the expression $(x+2y)^9$ is $84$ $672$ $8$ none of these
The coefficient of $x^6y^3$ in the expression $(x+2y)^9$ is$84$$672$$8$none of these
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55
ISI2018-DCG-4
The number of terms with integral coefficients in the expansion of $\left(17^\frac{1}{3}+19^\frac{1}{2}x\right)^{600}$ is $99$ $100$ $101$ $102$
The number of terms with integral coefficients in the expansion of $\left(17^\frac{1}{3}+19^\frac{1}{2}x\right)^{600}$ is$99$$100$$101$$102$
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56
ISI2018-DCG-17
The value of $^{13}C_{3} + ^{13}C_{5} + ^{13}C_{7} +\dots + ^{13}C_{13}$ is $4096$ $4083$ $2^{13}-1$ $2^{12}-1$
The value of $^{13}C_{3} + ^{13}C_{5} + ^{13}C_{7} +\dots + ^{13}C_{13}$ is$4096$$4083$$2^{13}-1$$2^{12}-1$
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57
ISI2016-MMA-14
The number of terms independent of $x$ in the binomial expansion of $\left(3x^2 + \dfrac{1}{x}\right)^{10}$ is $0$ $1$ $2$ $5$
The number of terms independent of $x$ in the binomial expansion of $\left(3x^2 + \dfrac{1}{x}\right)^{10}$ is $0$$1$$2$$5$
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59
Test-Book Live 2017
Which power of x has the greatest coefficient in the expansion of (1+1/2 x)^10 ?
Which power of x has the greatest coefficient in the expansion of (1+1/2 x)^10 ?
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