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Syllabus: Combinatorics: Counting, Recurrence relations, Generating functions.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}& \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{2020}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} & 1&1&0&0&2&1&0&0&1&0&0&0.6&2
\\\hline\textbf{2 Marks Count} & 2 &0&1&1&0&1&0&1&2&1&0&0.9&2
\\\hline\textbf{Total Marks} & 5 &1&2&2&2&3&0&2&5&2&0&2.4&5\\\hline
\end{array}}}$$

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Recent questions in Combinatory

1 votes
1 answer
631
ISI2018-DCG-12
Let $A=\{10,11,12,13, \dots ,99\}$. How many pairs of numbers $x$ and $y$ are possible so that $x+y\geq 100$ and $x$ and $y$ belong to $A$? $2405$ $2455$ $1200$ $1230$
Let $A=\{10,11,12,13, \dots ,99\}$. How many pairs of numbers $x$ and $y$ are possible so that $x+y\geq 100$ and $x$ and $y$ belong to $A$?$2405$$2455$$1200$$1230$
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1 votes
1 answer
632
ISI2018-DCG-14
In a room there are $8$ men, numbered $1,2, \dots ,8$. These men have to be divided into $4$ teams in such a way that every team has exactly $2$ ... of such $4$-team combinations is $\frac{8!}{2^4}$ $\frac{8!}{2^4(4!)}$ $\frac{8!}{4!}$ $\frac{8!}{(4!)^2}$
In a room there are $8$ men, numbered $1,2, \dots ,8$. These men have to be divided into $4$ teams in such a way thatevery team has exactly $2$ members, andthere are no ...
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2 votes
1 answer
633
ISI2018-DCG-17
The value of $^{13}C_{3} + ^{13}C_{5} + ^{13}C_{7} +\dots + ^{13}C_{13}$ is $4096$ $4083$ $2^{13}-1$ $2^{12}-1$
The value of $^{13}C_{3} + ^{13}C_{5} + ^{13}C_{7} +\dots + ^{13}C_{13}$ is$4096$$4083$$2^{13}-1$$2^{12}-1$
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1 votes
2 answers
637
CMI2018-A-5
How many paths are there in the plane from $(0,0)$ to $(m,n)\in \mathbb{N}\times \mathbb{N},$ if the possible steps from $(i,j)$ are either $(i+1,j)$ or $(i,j+1)?$ $\binom{2m}{n}$ $\binom{m}{n}$ $\binom{m+n}{n}$ $m^{n}$
How many paths are there in the plane from $(0,0)$ to $(m,n)\in \mathbb{N}\times \mathbb{N},$ if the possible steps from $(i,j)$ are either $(i+1,j)$ or $(i,j+1)?$$\binom...
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2 votes
2 answers
639
UGC NET CSE | June 2019 | Part 2 | Question: 2
How many ways are there to place $8$ indistinguishable balls into four distinguishable bins? $70$ $165$ $^8C_4$ $^8P_4$
How many ways are there to place $8$ indistinguishable balls into four distinguishable bins?$70$$165$$^8C_4$$^8P_4$
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3 votes
2 answers
640
UGC NET CSE | June 2019 | Part 2 | Question: 3
How many bit strings of length ten either start with a $1$ bit or end with two bits $00$ ? $320$ $480$ $640$ $768$
How many bit strings of length ten either start with a $1$ bit or end with two bits $00$ ?$320$$480$$640$$768$
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4 votes
1 answer
641
UGC NET CSE | June 2019 | Part 2 | Question: 7
How many cards must be selected from a standard deck of $52$ cards to guarantee that at least three hearts are present among them? $9$ $13$ $17$ $42$
How many cards must be selected from a standard deck of $52$ cards to guarantee that at least three hearts are present among them?$9$$13$$17$$42$
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2 votes
1 answer
642
UGC NET CSE | June 2019 | Part 2 | Question: 63
Consider the Euler’s phi function given by $\phi(n) = n \underset{p/n}{\Pi } \bigg( 1 – \frac{1}{p} \bigg)$ where $p$ runs over all the primes dividing $n$. What is the value of $\phi(45)$? $3$ $12$ $6$ $24$
Consider the Euler’s phi function given by$$\phi(n) = n \underset{p/n}{\Pi } \bigg( 1 – \frac{1}{p} \bigg)$$where $p$ runs over all the primes dividing $n$. What is t...
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1 votes
1 answer
643
#ACE ACADEMY BOOKLET QUESTION
The solution of $\sqrt{a_n} – 2\sqrt{a_{n-1}} + \sqrt{a_{n-2}} = 0$ where $a_0 = 1$ and $a_1 = 2$ is ${\Big[\frac{2^{n+1} + (-1)^n}{3}\Big]}^2$ $(n+1)^2$ $(n-1)^3$ $(n-1)^2$
The solution of $\sqrt{a_n} – 2\sqrt{a_{n-1}} + \sqrt{a_{n-2}} = 0$ where $a_0 = 1$ and $a_1 = 2$ is${\Big[\frac{2^{n+1} + (-1)^n}{3}\Big]}^2$$(n+1)^2$$(n-1)^3$$(n-1)^2...
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0 votes
0 answers
644
#Rosen exercise-1 ,question-71 counting
use mathematical induction to prove the sum rule for m tasks from the sum rule for two tasks.
use mathematical induction to prove the sum rule for m tasks from the sum rule for two tasks.
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0 votes
3 answers
645
Self Doubt-Combinatory
In how many ways we can put $n$ distinct balls in $k$ dintinct bins?? Will it be $n^{k}$ or $k^{n}$?? Taking example will be easy way to remove this doubt or some other ways possible??
In how many ways we can put $n$ distinct balls in $k$ dintinct bins??Will it be $n^{k}$ or $k^{n}$?? Taking example will be easy way to remove this doubt or some other wa...
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1 votes
2 answers
646
Rosen 7e Exercise-8.5 Question-15 page no-558 Inclusion-Exclusion
How many permutations of the 10 digits either begin with the 3 digits 987, contain the digits 45 in the fifth and sixth positions, or end with the 3 digits 123?
How many permutations of the 10 digits either begin with the 3 digits 987, contain the digits 45 in the fifth and sixth positions, or end with the 3 digits 123?
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1 votes
2 answers
647
Recurrence Relation Self-Doubt
What will be solution of recurrence relation if roots are like this: r1=-2, r2=2, r3=-2, r4=2 is this the case of repetitive roots?
What will be solution of recurrence relation if roots are like this: r1=-2, r2=2, r3=-2, r4=2is this the case of repetitive roots?
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0 votes
0 answers
649
Rosen 7e Exercise-8.2 Question no-23 page no-525 Recurrence Relation
Consider the nonhomogeneous linear recurrence relation $a_n$=$3a_{n-1}$+$2^n$ in the book solution is given $a_n$=$-2^{n+1}$ but I’m getting $a_n$=$3^{n+1}-2^{n+1}$
Consider the nonhomogeneous linear recurrence relation $a_n$=$3a_{n-1}$+$2^n$in the book solution is given $a_n$=$-2^{n+1}$but I’m getting $a_n$=$3^{n+1}-2^{n+1}$
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1 votes
2 answers
650
#probability(self doubt)
An automobile showroom has 10 cars, 2 of which are defective. If you are going to buy the 6th car sold that day at random, then the probability of selecting a defective car is??
An automobile showroom has 10 cars, 2 of which are defective. If you are going to buy the 6th car sold that day at random, then the probability of selecting a defective c...
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