The number of seven digit integer is to be formed using only 1, 2 &3 such that the sum of all the digits is 10 . so how many such seven digit number is possible ?

1 Answer
Apr 8, 2017

Answer:

I got #42# different numbers.

Explanation:

We start by listing the combinations of numbers that give us a digit sum of #10#.

We can have

#1, 1, 1, 1, 1, 3, 2#

And the number of different arrangements here is #(7!)/(5!) = 42#.

Is the above sequence the only possible?

Note that you need a certain number of #1#'s to make the number of digits #7# and the sum #10#, because the number #2222222# has a digit sum of #14#, for example. If we try other sequences, such as

#1, 1, 1, 1, 1, 1, 1, 3#

We either get sequences that are more or less than #7# terms or that have a sum other than #10#. I'm not sure how to prove that the above sequence is the only one possible, so I'll leave that to other contributors.

Hopefully this helps!