Question:
The total number of 3 -digit numbers, whose sum of digits is 10 , is
Solution:
Let three digit number is $x y z$
$x+y+z=10 ; \quad x \geq 1, y \geq 0 \quad z \geq 0$ ...(1)
Let $\mathrm{T}=\mathrm{x}-1 \Rightarrow \mathrm{x}=\mathrm{T}+1$ where $\mathrm{T} \geq 0$
Put in (1)
$\mathrm{T}+\mathrm{y}+\mathrm{z}=9 ; \quad 0 \leq \mathrm{T} \leq 8,0 \leq \mathrm{y}, \mathrm{z} \leq 9$
No. of non negative integral solution
$={ }^{9+3-1} \mathrm{C}_{3-1}-1($ when $\mathrm{T}=9)$
$=55-1=54$
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