Chapter 11.1, Problem 55PPSE

Solution

a.

Toidentify:The number of ways to choose three flags from a collection of seven different flags.

The total number of ways are 35.

Given information:

There are seven flags and need to select only three of them.

Explanation:

Consider the given information.

The total number of flags are 7 out of which 3 should be taken at a time.

Put $n=7\text{ and }r=3$ into the formula ${}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}$

  $\begin{array}{c}{}_7{C}_3=\frac{7!}{3!\left(7-3\right)!}\\ =\frac{7!}{3!\left(4\right)!}\\ =\frac{5\times 6\times 7}{6}\\ =35\end{array}$

Thus, the total number of ways are 35.

b.

Toidentify:The number ofdifferent orders to arrange three flags.

The possible arrangements are 210.

Given information:

The number of flags is 3.

Explanation:

Consider the given information.

The total number of flags are 7. 3 flags should be chosen and then arrange them.

Put $n=7\text{ and }r=3$ into the formula ${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

  $\begin{array}{c}{}_7{P}_3=\frac{7!}{\left(7-3\right)!}\\ =\frac{7!}{\left(4\right)!}\\ =5\times 6\times 7\\ =210\end{array}$

Thus, the possible arrangements are 210.

c.

Toexplain:How it can be used ${}_7{C}_3\cdot 3!$ to create the permutation formula. If they want to arrange three flags from a group of seven.

Multiply the number of combinations with 3! It will give the permutation.

Given information:

They want to arrange three flags from a group of seven.

Explanation:

Consider the given information.

From part a, it is known that ${}_7{C}_3=35$

Now there are 3 flags, if someone wants to arrange them then their position is important.

3 things can be arranged in 3! Ways.

Thus,

  $\begin{array}{r}{}_7{C}_3\cdot 3=35\cdot 3\\ =210\end{array}$