Question

The number of numbers between 100 to 1000 such that sum of their digits is 14, is _____.

Answer 1

Correct answer: 70

Number in this range will be 3-digit number. 

N = \(\overline{abc}\) = such that a + b + c = 14

Also, a ≥ 1, a, b, c ∈ {0, 1, 2,....9} 

Case I

All 3-digit same

⇒ 3a = 14 not possible

Case II

Exactly 2 digit same:

⇒ 2a + c = 14

(a, c) ∈ {(3, 8), (4, 6), (5, 4), (6, 2), (7, 0)}

⇒ \(\left( \frac{3!}{2!}\right)\) ways

⇒ 5 × 3 – 1

= 15 – 1 = 14

Case III

All digits are distinct a + b + c = 14

without losing generality a > b > c

⇒ 8 × 3! + 2(3! – 2!) = 48 + 8 = 56

= 0 + 14 + 56 = 70

Answer 2

The question can also be done by coefficient method as illustrated by the image