JEE PYQ: Permutation And Combination Question 18
Question 18 - 2020 (03 Sep Shift 2)
The total number of 3-digit numbers, whose sum of digits is 10, is ______.
Show Answer
Answer: 54
Solution
$x + y + z = 10$, $x \geq 1$, $y, z \geq 0$, $x \leq 9$, $y \leq 9$, $z \leq 9$. Let $t = x-1$: $t + y + z = 9$, $0 \leq t,y,z \leq 8$. Total without upper bound: ${}^{11}C_2 = 55$. Subtract case $t = 9$ (impossible since $\leq 8$): only when one variable $= 9$, others $= 0$, that’s for $t$: impossible. Actually for $t=9$: $y=z=0$, 1 case. So $55 - 1 = 54$.
BETA
