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Chapter 11: Problem 53

In how many ways can the digits in the number \(5,446,666\) be arranged?

Short Answer

Expert verified
The digits in the number '5,446,666' can be arranged in 420 different ways.

Step by step solution

01

Identify distinct digits and their counts

The number '5,446,666' contains four distinct digits: 5, 4, 6, and comma. Count their occurrences. There is one '5', two '4's, three '6's and one comma.
02

Use the permutation of multiset formula

To find out the number of distinct sequences that can be obtained by permuting all elements of the multiset, use the permutation of multiset formula. The formula is as follows: \[P_m = \frac{n!}{r_1! r_2! ... r_k!}\]Where \(n\) is the total number of elements, \(r_1, r_2, ..., r_k\) are the number of each type of item inside the multiset.
03

Calculate

Here \(n = 7\) (total number of elements including comma), \(r_1 = 1\) (number of '5's), \(r_2 = 2\) (number of '4's), \(r_3 = 3\) (number of '6's), and \(r_4 = 1\) (number of commas). Substitute the values into the formula and calculate it: \[P_m = \frac{7!}{1! * 2! * 3! * 1!} = \frac{5040}{12} = 420\]

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