Question

in how many ways can the digits in the number 9,551,111 be arranged?
□ ways
(type a whole number.)

Explanation:

Step1: Count the total number of digits

The number \(9,551,111\) has \(7\) digits.

Step2: Count the number of repeated digits

The digit \(5\) is repeated \(2\) times and the digit \(1\) is repeated \(4\) times.

Step3: Use the permutation formula for repeated elements

The formula for permutations of \(n\) objects where \(n_1\) of one kind, \(n_2\) of another kind, \(\cdots\), \(n_k\) of the \(k^{th}\) kind is \(\frac{n!}{n_1!n_2!\cdots n_k!}\). Here \(n = 7\), \(n_1=2\) (for digit \(5\)) and \(n_2 = 4\) (for digit \(1\)).
So the number of arrangements is \(\frac{7!}{2!4!}\).
Calculate \(7! = 7\times6\times5\times4\times3\times2\times1=5040\), \(2! = 2\times1 = 2\), \(4! = 4\times3\times2\times1=24\).
Then \(\frac{5040}{2\times24}=\frac{5040}{48}=105\).

Answer:

\(105\)