Counting sets How many even 11digit numbers no leading zeros

(Counting sets)

How many even 11-digit numbers (no leading zeros) have at least three 3\'s?

Solution

Case 1. First digit is a 3 ie leading digit is a 3

So we can choose 2 more digits for 3s in C(10,2)=45 ways

Other 9 digits can be filled in 9^10 ways

So,

45*9^10 numbers

Case 2. First digit is not a 3

So 8 ways to fill first digit since it cannot be a 0 or 3

Choose 2 other places for 3 in 45 ways

Other 9 places are filled in 9^10 ways

So,

8*45*9^10 numbers

So total

9*45*9^10 numbers