Question

A closet has a \[5\] pair of shoes. The number of ways in which \[4\] shoes can be drawn from it such that there will be no complete pair is?
A.\[80\]
B.\[160\]
C.\[200\]
D.None of the foregoing numbers.

Answer 1

Hint: In this problem shoes are selected without following any order so we will use here the combination formula. Permutation is represented by \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\] and combination is represented by
 $ {}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} $ where, $ r $ is combination of objects from a set of $ n $ objects.

Complete step-by-step answer:
We are given that:
Total shoes \[ = {\text{ }}5\] pairs of shoes
For selection, we use combinations.
So now we know total ways of selecting \[4\] pair out of \[5\] \[ = {\text{ }}{(^5}{C_4})\]
As there are \[4\] pairs,
Now for every pair, on selecting \[1\]shoe, this can be done in $ {{(^2}{C_1})^4} $ ways.
Therefore total no. of ways \[ = {(^5}{C_4}){{(^2}{C_1})^4}\]
 $
   \Rightarrow \left( {\dfrac{{5!}}{{4!\left( {5 - 4} \right)!}}} \right){\left( {\dfrac{{2!}}{{1!\left( {2 - 1} \right)!}}} \right)^4} \\
   \Rightarrow 5 \times {2^4} \\
   \Rightarrow 5 \times 16 \\
   \Rightarrow 80 \;
 $
So, therefore, there are \[80\]ways.
So, the correct answer is “Option A”.

Note: Students should remember that selecting without any order is an example of combinations. The difference between permutations and combinations is that Permutation is a selection of objects in which order of the objects matters and Combination is the number of possible combinations of $ r $ objects from a set of $ n $ objects.