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

Apr 25, 2018

$\text{630}$

#### Explanation:

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

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

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

$\text{possibilities of arranging them.}$

$\text{So we need to count how many times the items occur :}$
$\text{Here we have 7 items : two 579 and one 6, so}$

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

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