Question

In how many ways can the digits in the number 6759957 be arranged?

Answer 1

`"630"`

Explanation:

`(7!)/((2!)^3) = 630`

`"In general when we arrange n items, where there are k different"`
`"items that occur each "n_i" times, for "i=1,2,...,k", then we"`
`"have"`

`(n!)/((n_1)!(n_2)!...(n_k)!)`

`"possibilities of arranging them."`

`"So we need to count how many times the items occur :"`
`"Here we have 7 items : two 579 and one 6, so"`

`(7!)/(2!2!2!1!) = 630 " possibilities"`

`"This is called a multinomial coefficient."`
`"The philosophy behind it is simple. We would have n! ways of"`
`"arranging them if they were different, but the identical items"`
`"can be arranged in "n_i!" ways, without affecting the outcome"`
`"so we divide through all the "(n_i)!.`