Question

A rack has 5 different pairs of shoes. The number of ways in which 4 shoes can be chosen from it. so that there will be no complete pair, is

A 1920
B 200
C 110
D 80​

Answer 1

The number of ways to choose 4 shoes from 5 pairs without completing a pair is 80, which is found by combining 5 ways to select the pairs and 16 ways to choose individual shoes within those pairs.

Step-by-step explanation:

The question involves calculating the number of ways to choose 4 shoes from 5 pairs without completing a pair. To find the solution, we need to break down the problem into two steps:

1. Select 4 different pairs from the 5 pairs available. This can be done in 5 choose 4 ways, which is equal to 5 ways since there are 5 pairs and we want to select any 4 of them.
2. From each of the selected 4 pairs, choose one shoe. This can be done in 2 ways per pair since each pair consists of 2 shoes. As we have 4 pairs, the total number of ways is 2^4 which equals 16 ways.

Multiplying the number of ways to choose the pairs by the number of ways to choose individual shoes gives us the total number of ways: 5 ways to choose the pairs multiplied by 16 ways to choose the shoes, which equates to 80 ways.

Hence, the correct answer is 80, which is option D.