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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

3 votes
1 answer
1421
33. How many different strings can be made from the letters in ORONO, using some or all of the letters?
The answer is 63.But I got 57.Approch:1 letters: = O,R,N(=3)2 letters: = OO,OR,ON,NR(1+2+2+2=7)3 Letters: = OOO,OOR,OON,ORN(1+3+3+6 = 13)4 Letters : OOOR,OOON,OORN(4+4+6 ...
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0 votes
0 answers
1422
Prove the hockeystick identity using a combinatorial argument.
Here's Hockey Stick Identity: what is Hockey Stick Identity means and Where it is used in Practical Application? Also Prove Suitable Explanation of the combinatorial argument.
Here's Hockey Stick Identity:what is Hockey Stick Identity means and Where it is used in Practical Application?Also Prove Suitable Explanation of the combinatorial argume...
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8 votes
1 answer
1423
There are six runners in the 100-yard dash. How many ways are there for three medals to be awarded if ties are possible?
There are six runners in the 100-yard dash. How many ways are there for three medals to be awarded if ties are possible? (The runner or runners who finish with the fastes...
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6 votes
1 answer
1425
The English alphabet contains 21 consonants and five vowels. How many strings of six lowercase letters of the English alphabet contain a) exactly one vowel? b) exactly two vowels? c) at least one vowel? d) at least two vowels?
Hi Answer to each option is given as:a) 122,523,030b) 72,930,375c) 223,149,655d) 100,626,625And I used the following approach to each option but answers don't match.a) C(...
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43 votes
11 answers
1428
GATE IT 2004 | Question: 35
In how many ways can we distribute $5$ distinct balls, $B_1, B_2, \ldots, B_5$ in $5$ distinct cells, $C_1, C_2, \ldots, C_5$ such that Ball $B_i$ is not in cell $C_i$, $\forall i= 1,2,\ldots 5$ and each cell contains exactly one ball? $44$ $96$ $120$ $3125$
In how many ways can we distribute $5$ distinct balls, $B_1, B_2, \ldots, B_5$ in $5$ distinct cells, $C_1, C_2, \ldots, C_5$ such that Ball $B_i$ is not in cell $C_i$, $...
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41 votes
1 answer
1429
GATE IT 2004 | Question: 34
Let $H_1, H_2, H_3,$ ... be harmonic numbers. Then, for $n \in Z^+$, $\sum_{j=1}^{n} H_j$ can be expressed as $nH_{n+1} - (n + 1)$ $(n + 1)H_n - n$ $nH_n - n$ $(n + 1) H_{n+1} - (n + 1)$
Let $H_1, H_2, H_3,$ ... be harmonic numbers. Then, for $n \in Z^+$, $\sum_{j=1}^{n} H_j$ can be expressed as$nH_{n+1} - (n + 1)$$(n + 1)H_n - n$$nH_n - n$$(n + 1) H_{n+...
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38 votes
3 answers
1431
GATE IT 2008 | Question: 25
In how many ways can $b$ blue balls and $r$ red balls be distributed in $n$ distinct boxes? $\frac{(n+b-1)!\,(n+r-1)!}{(n-1)!\,b!\,(n-1)!\,r!}$ $\frac{(n+(b+r)-1)!}{(n-1)!\,(n-1)!\,(b+r)!}$ $\frac{n!}{b!\,r!}$ $\frac{(n + (b + r) - 1)!} {n!\,(b + r - 1)}$
In how many ways can $b$ blue balls and $r$ red balls be distributed in $n$ distinct boxes?$\frac{(n+b-1)!\,(n+r-1)!}{(n-1)!\,b!\,(n-1)!\,r!}$$\frac{(n+(b+r)-1)!}{(n-1)!\...
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6 votes
1 answer
1432
GATE CSE 1996 | Question: 9
The Fibonacci sequence $\{f_1, f_2, f_3 \ldots f_n\}$ is defined by the following recurrence:$f_{n+2} = f_{n+1} + f_n, n \geq 1; f_2 =1:f_1=1$Prove by induction that every third element of the sequence is even.
The Fibonacci sequence $\{f_1, f_2, f_3 \ldots f_n\}$ is defined by the following recurrence:$$f_{n+2} = f_{n+1} + f_n, n \geq 1; f_2 =1:f_1=1$$Prove by induction that ev...
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18 votes
2 answers
1433
GATE CSE 1994 | Question: 15
Use the patterns given to prove that $\sum\limits_{i=0}^{n-1} (2i+1) = n^2$ (You are not permitted to employ induction) Use the result obtained in (A) to prove that $\sum\limits_{i=1}^{n} i = \frac{n(n+1)}{2}$
Use the patterns given to prove that$\sum\limits_{i=0}^{n-1} (2i+1) = n^2$(You are not permitted to employ induction)Use the result obtained in (A) to prove that $\sum\li...
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26 votes
5 answers
1434
GATE CSE 1994 | Question: 1.15
The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is $n$ $n^2$ $\frac{n(n-1)}{2}$ $\frac{n(n+1)}{2}$
The number of substrings (of all lengths inclusive) that can be formed from a character string of length $n$ is$n$$n^2$$\frac{n(n-1)}{2}$$\frac{n(n+1)}{2}$
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23 votes
3 answers
1436
GATE CSE 1998 | Question: 1.23
How many sub strings of different lengths (non-zero) can be formed from a character string of length $n$? $n$ $n^2$ $2^n$ $\frac{n(n+1)}{2}$
How many sub strings of different lengths (non-zero) can be formed from a character string of length $n$?$n$$n^2$$2^n$$\frac{n(n+1)}{2}$
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38 votes
7 answers
1437
GATE CSE 1999 | Question: 2.2
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves? $1638$ $2100$ $2640$ None of the above
Two girls have picked $10$ roses, $15$ sunflowers and $15$ daffodils. What is the number of ways they can divide the flowers among themselves?$1638$$2100$$2640$None of th...
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30 votes
5 answers
1438
GATE CSE 1999 | Question: 1.3
The number of binary strings of $n$ zeros and $k$ ones in which no two ones are adjacent is $^{n-1}C_k$ $^nC_k$ $^nC_{k+1}$ None of the above
The number of binary strings of $n$ zeros and $k$ ones in which no two ones are adjacent is$^{n-1}C_k$$^nC_k$$^nC_{k+1}$None of the above
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28 votes
8 answers
1440
GATE CSE 2005 | Question: 50
Let $G(x) = \frac{1}{(1-x)^2} = \sum\limits_{i=0}^\infty g(i)x^i$, where $|x| < 1$. What is $g(i)$? $i$ $i+1$ $2i$ $2^i$
Let $G(x) = \frac{1}{(1-x)^2} = \sum\limits_{i=0}^\infty g(i)x^i$, where $|x| < 1$. What is $g(i)$?$i$$i+1$$2i$$2^i$
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