The total number of 3 -digit numbers,
Question:

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

Solution:

Let $x y z$ be the three digit number

$x+y+z=10, x \leq 1, y \geq 0, z \geq 0$

$x-1=t \Rightarrow x=1+t$      $x-1 \geq 0, t \geq 0$

$t+y+z=10-1=9$      $0 \leq t, z, z \leq 9$

$\therefore$ Total number of non-negative integral solution

$={ }^{9+3-1} C_{3-1}={ }^{11} C_{2}=\frac{11 \cdot 10}{2}=55$

But for $t=9, x=10$, so required number of integers

$=55-1=54$

 

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