Question

How many 3 digits numbers have the sum of their digits as 15?

• A. 62
• B. 79
• C. 69
• D. 56

Answer 1

The correct option is C 69

a + b + c = 15
but a > 0
So, a'+ b + c = 14
Number of ways: ${}^{16}{C}_2$
Here, when a' is 9, a>9 (invalid case)
put a' = x + 9
x + 9 + b + c = 14
x + b + c = 5
This can be done in ${}^7{C}_2$ ways
When b > 9, it is an invalid case
put b = y + 10
a' + y + 10 + c = 14
a' + y + c = 4
This can be done in ${}^6{C}_2$ ways similarly, for cases where c>9, there are ${}^6{C}_2$ ways therefore, total count of 3 digits numbers which have sum of its digit as 15 are: ${}^{16}{C}_2-({}^7{C}_2+{}^6{C}_2+{}^6{C}_2)=120-(21+15+15)$
$\to$ 120 - 51 = 69