Question

The number of seven digit i ntegers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is
(a) 55
(b) 66
(c) 77
(d) 88

Answer 1

There are only two integer partitions of 10 of length 7 formed from the digits 1,2,3: {3,2,1,1,1,1,1} and {2,2,2,1,1,1,1}. The number of ways you can form numbers from the first set can be computed using a multinomial:

7!1!1!5!=42

and for the second set is

7!3!4!=35

so the total is 77.

Confirmation using Mathematica:

Length@Select[Tuples[{1, 2, 3}, 7], Total[#] == 10 &]

(* 77 *)