Question

A classic counting problem is to determine the number of different ways that the letters of "balloon" can be arranged. Find that number.

Answer 1

`\bf \stackrel{permutations}{_nP_r}=\cfrac{n!}{(n-r)!}\qquad\qquad \qquad \stackrel{balloon}{_7P_1}=\cfrac{7!}{(7-1)!}`

Answer 2

The number is:

1260

Explanation:

We know that the number of ways of arranging n items is calculated by the method of permutation.

If n letters are to be arranged such that there are
`r_1,r_2` items each of the same type.

Then, the number of ways of arranging is:

`(n!)/(r_1!* r_2!)`

We are asked to find the number of ways of arranging the letters of "Balloon"

There are a total of 7 words such that 'l' occurs two times and 'o' occurs two times.

Hence, the number of ways of arranging them are:

`=(7!)/(2!* 2!)\\\\=(7* 6* 5* 4* 3* 2!)/(2!* 2)\\\\=1260\ ways`