Question

examples 2 and 3

1. phone numbers what is the probability that a 7 - digit telephone number generated using the digits 2, 3, 2, 5, 2, 7, and 3 is the number 222 - 3357?

Explanation:

Step1: Find total permutations

We have 7 digits with duplicates: 2 appears 3 times, 3 appears 2 times, and 5, 7 once each. The formula for permutations of multi - set is $\frac{n!}{n_1!n_2!\cdots n_k!}$, where $n = 7$, $n_1=3$ (for 2s), $n_2 = 2$ (for 3s). So total permutations $=\frac{7!}{3!2!}=\frac{7\times6\times5\times4\times3!}{3!\times2\times1}=420$.

Step2: Determine favorable outcomes

There is only 1 favorable outcome (the number 222 - 3357).

Step3: Calculate probability

Probability $=\frac{\text{favorable outcomes}}{\text{total outcomes}}=\frac{1}{420}$.

Answer:

$\frac{1}{420}$