JEE MAIN Permutations And Combinations Previous Year Questions (PYQs) – Page 1 of 9
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If ${}^n{P_r} = {}^n{P_{r + 1}}$ and ${}^n{C_r} = {}^n{C_{r - 1}}$, then the value of r is equal to :
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Solution
Let $a \in \left(0, \dfrac{\pi}{2}\right)$ be fixed.
If
$\displaystyle \int \dfrac{\tan x + \tan a}{\tan x - \tan a} , dx = A(x)\cos 2a + B(x)\sin 2a + C,$
where $C$ is a constant of integration,
then the functions $A(x)$ and $B(x)$ are respectively:
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Solution
Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?
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Solution
Let there be two families with 3 members each (say Family A and Family B), and one family with 4 members (Family C).
Step 1: Since members of the same family must sit together, treat each family as a single block.
Thus, there are 3 blocks: A, B, and C.
They can be arranged in $3! = 6$ ways.
Step 2: Now arrange members within each family:
Family A (3 members): $3!$ ways
Family B (3 members): $3!$ ways
Family C (4 members): $4!$ ways
Step 3: Total number of arrangements = $3! \times 3! \times 3! \times 4!$
Step 4: Simplify:
$3! = 6$ and $4! = 24$
$\Rightarrow (3!)^3\times4!$
∴ Total number of ways = $\boxed{5184}$
Step 1: Since members of the same family must sit together, treat each family as a single block.
Thus, there are 3 blocks: A, B, and C.
They can be arranged in $3! = 6$ ways.
Step 2: Now arrange members within each family:
Family A (3 members): $3!$ ways
Family B (3 members): $3!$ ways
Family C (4 members): $4!$ ways
Step 3: Total number of arrangements = $3! \times 3! \times 3! \times 4!$
Step 4: Simplify:
$3! = 6$ and $4! = 24$
$\Rightarrow (3!)^3\times4!$
∴ Total number of ways = $\boxed{5184}$
A group of students comprises of $5$ boys and $n$ girls. If the number of ways in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team is $1750$, then $n$ is equal to:
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Solution
Let A and B be two sets containing 2 elements and 4 elements respectively. The
number of subsets of AXB having 3 or more elements is :
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Solution
n–digit numbers are formed using only three digits 2,5,7.
The smallest value of n for which 900 such distinct numbers can be formed is :
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Solution
From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include atleast 4 batsmen and atleast 4 bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is
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Solution
Let $A={1,3,7,9,11}$ and $B={2,4,5,7,8,10,12}$. The total number of one–one maps $f:A\to B$ such that $f(1)+f(3)=14$ is:
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Solution
Let $T_{n}$ be the number of all possible triangles formed by joining vertices
of an $n$-sided regular polygon. If $T_{n+1}-T_{n}=10$, then the value of $n$ is :
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Solution
All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is :
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Solution
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