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

Answer:

                 The number is:

                          1260

Step-by-step 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:

                   $\dfrac{n!}{r_1!\times r_2!}$

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:

     $=\dfrac{7!}{2!\times 2!}\\\\=\dfrac{7\times 6\times 5\times 4\times 3\times 2!}{2!\times 2}\\\\=1260\ ways$