Chapter 13.9, Problem 36HP

To determine

To explain: value of $P\left(x,y\right)$ sometimes, always or never greater than the value of $C\left(x,y\right)$ .

Answer to Problem 36HP

  $P\left(x,y\right)$ is greater than the value of $C\left(x,y\right)$ when repetition is not allowed.

Explanation of Solution

Given information:

  $x$ and $y$ are positive integers and $x\ge y$ .

Permutation is an arrangement and thus, order is important.

In combination order is important.

Consider the case when repetition is not allowed.

Two roses, one red and other yellow in color.

Holding red rose in left hand and yellow rose in right hand is an arrangement of holding the roses.

So, one permutation.

Other permutation is holding yellow colored rose in left hand and red colored rose in right hand but one red and one yellow irrespective of holding position is combination.

Therefore, the number of permutations is twice as great as the number of combinations.

Also consider the case when repetition is not allowed.

In that case, there is red rose in left hand as well as in right hand.

That gives one permutation and one combination.

That is why, in case when repetition is allowed, the number of permutations and combinations are similar.