JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Permutations and Combinations

The total number of 3 - digit numbers, whose sum of digits is 10 , is ______.

Answer: 54

Solution

Let the number be

\overline{x y z}

with

x \geq 1

and

y, z \geq 0

and

x + y + z = 10

. Put

x = 1 + t

with

t \geq 0

, so

t + y + z = 9

; the non-negative solutions number

(2 9 + 2) = (2 11) = 55

. One of these,

t = 9

with

y = z = 0

, forces

x = 10

, which is not a single digit and must be dropped. That leaves

55 - 1 = 54

such numbers.

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Integer answer