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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}& \textbf{2024-1} & \textbf{2024-2} & \textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &0&0&1& 1&1&0&0&0.5&1
\\\hline\textbf{2 Marks Count} &0&0&1& 2 &0&1&0&0.67&2
\\\hline\textbf{Total Marks} & 0&0&3&5 &1&2&0&1.83&5\\\hline
\end{array}}}$$

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Most answered questions in Combinatory

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162
Kenneth Rosen Edition 7 Exercise 6.5 Question 7 (Page No. 432)
How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?
How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?
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2 answers
163
Kenneth Rosen Edition 7 Exercise 6.4 Question 25 (Page No. 422)
Let n be a positive integer. Show that $\binom{2n}{n + 1} + \binom{2n}{n} = \dfrac{\binom{2n + 2}{n + 1}}{2}.$
Let n be a positive integer. Show that $\binom{2n}{n + 1} + \binom{2n}{n} = \dfrac{\binom{2n + 2}{n + 1}}{2}.$
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2 answers
165
Kenneth Rosen Edition 7 Exercise 6.2 Question 20 (Page No. 406)
Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in the sequence $22, 5, 7, 2, 23, 10, 15, 21, 3, 17.$
Find an increasing subsequence of maximal length and a decreasing subsequence of maximal length in the sequence $22, 5, 7, 2, 23, 10, 15, 21, 3, 17.$
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167
Kenneth Rosen Edition 7 Exercise 6.1 Question 50 (Page No. 398)
How many bit strings of length $10$ contain either five consecutive $0s$ or five consecutive $1s?$
How many bit strings of length $10$ contain either five consecutive $0s$ or five consecutive $1s?$
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168
Kenneth Rosen Edition 7 Exercise 6.1 Question 25 (Page No. 397)
How many strings of three decimal digits do not contain the same digit three times? begin with an odd digit? have exactly two digits that are $4s?$
How many strings of three decimal digitsdo not contain the same digit three times?begin with an odd digit?have exactly two digits that are $4s?$
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2 votes
2 answers
170
Kenneth Rosen Edition 7 Exercise 6.1 Question 21 (Page No. 396)
How many positive integers between $50$ and $100$ are divisible by $7?$ Which integers are these? are divisible by $11?$ Which integers are these? are divisible by both $7$ and $11?$ Which integers are these?
How many positive integers between $50$ and $100$are divisible by $7?$ Which integers are these?are divisible by $11?$ Which integers are these?are divisible by both $7$ ...
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1 votes
2 answers
171
Kenneth Rosen Edition 7 Exercise 6.1 Question 20 (Page No. 396)
How many positive integers between $5$ and $31$ are divisible by $3?$ Which integers are these? are divisible by $4?$ Which integers are these? are divisible by $3$ and by $4?$ Which integers are these?
How many positive integers between $5$ and $31$ are divisible by $3?$ Which integers are these?are divisible by $4?$ Which integers are these?are divisible by $3$ and by ...
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172
Kenneth Rosen Edition 7 Exercise 6.1 Question 16 (Page No. 396)
How many strings are there of four lowercase letters that have the letter $x$ in them?
How many strings are there of four lowercase letters that have the letter $x$ in them?
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174
Kenneth Rosen Edition 7 Exercise 6.1 Question 9 (Page No. 396)
How many different three-letter initials are there that begin with an $A?$
How many different three-letter initials are there that begin with an $A?$
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175
Kenneth Rosen Edition 7 Exercise 6.1 Question 8 (Page No. 396)
How many different three-letter initials with none of the letters repeated can people have?
How many different three-letter initials with none of the letters repeated can people have?
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177
Kenneth Rosen Edition 7 Exercise 6.1 Question 6 (Page No. 396)
There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?
There are four major auto routes from Boston to Detroit and six from Detroit to Los Angeles. How many major auto routes are there from Boston to Los Angeles via Detroit?
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179
Kenneth Rosen Edition 7 Exercise 6.1 Question 4 (Page No. 396)
A particular brand of shirt comes in $12$ colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?
A particular brand of shirt comes in $12$ colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt a...
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