(D) The number of ways to arrange $r$ objects out of $n$ distinct objects is given by the permutation formula $^nP_r = \frac{n!}{(n-r)!}$.Here,$n = 5$

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(D) The number of ways to arrange $r$ objects out of $n$ distinct objects is given by the permutation formula $^nP_r = \frac{n!}{(n-r)!}$.Here,$n = 5$ (vacant seats) and $r = 3$ (passengers).Therefore,the number of ways is $^5P_3 = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2!}{2!} = 5 \times 4 \times 3 = 60$.

Class 11 Mathematics Permutation and Combination Definition of permutation, Number of permutations with or without repetition, Conditional permutations

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In a train,five seats are vacant. In how many ways can three passengers sit?

A
$20$
B
$30$
C
$10$
D
$60$

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Class 11

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Permutation and Combination

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Definition of permutation, Number of permutations with or without repetition, Conditional permutations

$10$ men and $6$ women are to be seated in a row so that no two women sit together. The number of ways they can be seated is:

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How many numbers lying between $10$ and $1000$ can be formed from the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ (repetition is allowed)?

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Out of $10$ persons,$A, B$,and $C$ are to speak at a function. If $A$ wants to speak before $B$ and $B$ wants to speak before $C$,in how many ways can they speak?

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Assuming that no two consecutive digits are same,the number of $n$-digit numbers is:

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If $0 < r < s < n$ and ${}^n P_r = {}^n P_s$,then find the value of $r + s$.

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In a train,five seats are vacant. In how many ways can three passengers sit?

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$10$ men and $6$ women are to be seated in a row so that no two women sit together. The number of ways they can be seated is:

How many numbers lying between $10$ and $1000$ can be formed from the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ (repetition is allowed)?

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