Chapter 12.6, Problem 4P

To determine

To calculate: The number of different ways we can choose to form a bouquet.

Answer to Problem 4P

The number of ways are $455$ .

Explanation of Solution

Given information:

3 types of flowers for a bouquet from a selection of 15 types of flowers.

Formula used:

A combination is a selection of objects without regard to order. The expression $C{}_r$ represents the number of combination of $n$ objects chosen $r$ at a time.

  $C{}_r=\frac{n!}{r!\left(n-r\right)!}$

For any positive integer $n$, the expression $n$ factorial is written as $n!$ and is the product of integers from n down to 1.

Calculation:

According to the question,

We want to make bouquet from 3 different types of flowers.

We have 5 types of flowers.

Therefore,

  $\begin{array}{l}C{}_3=\frac{15!}{3!\left(15-3\right)!}\\ =\frac{15!}{3!12!}\end{array}$

Now, solving the factorials .

  $\begin{array}{l}=\frac{15\times 14\times 13\times 12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\times 3\times 2\times 1}\\ =\frac{15\times 14\times 13}{3\times 2\times 1}\\ =5\times 7\times 13\\ =455\end{array}$

Hence, the number of ways are $455$ .