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  • The total number of 3-digit numbers, whose sum of digits is 10, is _______. Option: 1 54 Option: 2 28 Option: 3 64 Option: 4 32  

The total number of 3-digit numbers, whose sum of digits is 10, is _______.
Option: 1 54
Option: 2 28
Option: 3 64
Option: 4 32
 

Answers (1)

best_answer

Let three-digit number is xyz

\\x+y+z=10 ; x \geq 1, y \geq 0 z \geq 0\;\;\;\;\;\;\;\;\ldots(1)\\ Let \;\mathrm{T}=\mathrm{x}-1 \Rightarrow \mathrm{x}=\mathrm{T}+1 \;where \;\mathrm{T} \geq 0\\ Put \;in \;(1)\\ \mathrm{T}+\mathrm{y}+\mathrm{z}=9 ;\; 0 \leq \mathrm{T} \leq 8,0 \leq \mathrm{y}, \mathrm{z} \leq 9\\ \text{No. of non negative integral solution} =\\{ }^{9+3-1} \mathrm{C}_{3-1}-1 \quad($ when $\mathrm{T}=9)$ $=55-1=54$

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