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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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14 votes
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305
ISI 2004 MIII
The number of permutation of $\{1,2,3,4,5\}$ that keep at least one integer fixed is. $81$ $76$ $120$ $60$
The number of permutation of $\{1,2,3,4,5\}$ that keep at least one integer fixed is.$81$$76$$120$$60$
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306
Kenneth Rosen Edition 7 Exercise 6.5 Question 11 (Page No. 432)
How many ways are there to choose eight coins from a piggy bank containing $100$ identical pennies and $80$ identical nickels?
How many ways are there to choose eight coins from a piggy bank containing $100$ identical pennies and $80$ identical nickels?
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309
Kenneth Rosen Edition 7 Exercise 6.1 Question 14 (Page No. 396)
. How many bit strings of length $n,$ where n is a positive integer, start and end with $1s?$
. How many bit strings of length $n,$ where n is a positive integer, start and end with $1s?$
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310
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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311
Kenneth Rosen Edition 7 Exercise 6.5 Question 63 (Page No. 434)
Prove the Multinomial Theorem: If $n$ ... $C(n:n_{1},n_{2},\dots,n_{m}) = \dfrac{n!}{n_{1}!n_{2}!\dots n_{m}!}$ is a multinomial coefficient.
Prove the Multinomial Theorem: If $n$ is a positive integer, then $\displaystyle{}(x_{1} + x_{2} + \dots + x_{m})^{n} = \sum_{n_{1} + n_{2} + \dots + n_{m} = n}\:\: C(n:n...
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312
Kenneth Rosen Edition 7 Exercise 6.5 Question 8 (Page No. 432)
How many different ways are there to choose a dozen donuts from the $21$ varieties at a donut shop?
How many different ways are there to choose a dozen donuts from the $21$ varieties at a donut shop?
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314
Kenneth Rosen Edition 7 Exercise 8.3 Question 10 (Page No. 535)
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$
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316
Kenneth Rosen Edition 7 Exercise 6.5 Question 12 (Page No. 432)
How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a piggy bank contain if it has $20$ coins in it?
How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a piggy bank contain if it has $20$ coins in it?
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318
Kenneth Rosen Edition 7 Exercise 6.1 Question 61 (Page No. 398)
A wired equivalent privacy (WEP) key for a wireless fidelity (WiFi) network is a string of either $10, 26,\:\text{or}\: 58$ hexadecimal digits. How many different WEP keys are there?
A wired equivalent privacy (WEP) key for a wireless fidelity (WiFi) network is a string of either $10, 26,\:\text{or}\: 58$ hexadecimal digits. How many different WEP key...
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319
Kenneth Rosen Edition 7 Exercise 8.3 Question 9 (Page No. 535)
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find$f (5)$$f (125)$$f (3125)$
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320
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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