1 7 people go to a cinema and park cars outside which r identical While returning only 3 - Maths - Permutations and Combinations

Question

1 7 people go to a cinema and
park cars outside which r identical While returning only 3 of them
are able to unloack their cars Find the total number of ways in
which this can be done

Aakash Sharma · Accepted Answer

If n things are arranged in a row, then the number of ways in which they can be deranged so that none of then occupies its original place is

$n! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + ...... = (- 1)^{n} \frac{1}{n!})$

Number of identical cars = 7

Number of ways in which 3 people were able to unlock their cars = 1.

Number of ways in which remaining 4 people were not able to unlock their cars = $4! (1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!})$

$= 24 (1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24}) = 24 (\frac{12 - 4 + 1}{24}) = 9$

∴ Total number of ways = 1 × 9 = 9

1. 7 people go to a cinema and park cars outside which r identical. While returning only 3 of them are able to unloack their cars. Find the total number of ways in which this can be done.