Quote:

How many 7-digit even numbers less than 3,000,000 can be formed

using the following digits: 1, 2, 2, 3, 5, 5, 6 ?

I understand this part:

There are four cases:

1 is the first digit, 6 is the last digit << Case 1

2 is the first digit, 6 is the last digit << Case 2

2 is the first digit, 2 is the last digit << Case 3

1 is the first digit, 2 is the last digit << Case 4

But this is the part I don't understand . . .

Why is it that for case 1, the answer is: $\displaystyle \frac{5!}{2!\,2!}$ ?

How do we get the 5 and why do we use 5 and not 7?

I understand the denominator, since there are duplicates of 2 and 5,

but why 5 in the numerator and not 7?