Question

How many distinct seven-digit phone numbers can you create from the numbers 4, 4, 5, 6, 7, 7, 7? (e.g. 445-6777 is one example)?

Answer 1

You have 7 numbers, two numbers 4 , one 5, one 6 and three numbers 7.

From these 7 numbers you can create $7!=1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7=5040$ phone numbers, but they are not all different.

If you have two numbers 4, then you have to divide obtained number by 2! (the number of all possible permutations of two numbers 4) and since you have three numbers 7 you should divide the obtained number by 3! (the number of all possible permutations of three numbers 7). Then

$\dfrac{5040}{2!\cdot 3!}= \dfrac{5040}{1\cdot2\cdot1\cdot2\cdot3}=420$.

Answer: You can create 420 different phone numbers.