Question

For how many natural numbers less than 10^⁵ is the sum of their digits equal to 10?​

Answer 1

Answer:

Step-by-step explanation:

Given For how many natural numbers less than 10^⁵ is the sum of their digits equal to 10?

The total number of ways of dividing n among r so that each will receive 0,1,2 or more is given by     (n + r – 1) C r – 1

So x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x2 + x1 + x0 = 10

Now n = 10, r = 5

So we get 10 + 5 – 1 C 5 – 1

        = 14 C 4  

Answer 2

Answer:

14c4-5

Step-by-step explanation:

Take 10 balls each ball representing a value of 1. Sum of values of 10 balls = 10.
Take 4 lines that divide these 10 balls into 5 blocks

something like 00|000|0|00|00. This number would represent 23122.
These 4 lines and 10 balls can be arranged in A = 14!/(4! * 10!)
But these A arrangements also have in them 5 wrong arrangements, which are
||||0000000000 ->2 ways
|||0000000000| ->2 ways
||0000000000|| ->1 way

These are wrong because be cannot represent 10 balls = value 10 in a single digit.
Valid Arrangements are A-5 = 14c4-5 = 996

P.S : The above method is knows as Stars and Bars method of counting. Many resources are available on the internet.