The number of integers, between 100 and 1000 having the sum of their digits equals to 14 ,…

Question

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is \_\_\_\_\_\_\_\_\_\_.

Answer: 70

Accepted Answer

Number in this range will be 3-digit number.

$N=\vec{abc}$ such that $a+b+c=14$

Also, $a\ge 1,\, a,b,c\in {0,1,2,\ldots 9}$

Case I

All 3-digit same

$\Rightarrow 3a=14$ not possible

Case II

Exactly 2 digit same:

$\Rightarrow 2a+c=14$

$\begin{gathered} (a,c)\in {(3,8),(4,6),(5,4),(6,2),(7,0)} \ \Rightarrow (\frac{3!}{2!})\Rightarrow 5	imes 3-1 \ =15-1=14 \end{gathered}$

Case III

All digits are distinct

$a+b+c=14$

without losing generality $a>b>c$

$\begin{gathered} (a,b,c)\in \begin{gathered} (9,5,0),(9,4,1),(9,3,2) \ (8,6,0),(8,5,1),(8,4,2) \ (7,6,1),(7,5,2),(7,4,3) \ (6,5,3) \end{gathered} \ \Rightarrow 8	imes 3!+2(3!-2!)=48+8=56 \ =0+14+56=70 \end{gathered}$

Solutions · 1