(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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Permutation and Combination

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

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

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How many words of $4$ distinct letters can be formed using the letters of the word $DHOLPUR$ if $L$ and $P$ are always excluded?

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If $^n{P_r} = 720 \times {^n{C_r}}$,then $r$ is equal to:

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In how many ways $n$ books can be arranged in a row so that two specified books are not together?

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All the letters of the word $MOTHER$ are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word $MOTHER$ is

MediumTS EAMCET 2025

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

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

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