devarshi-dt-logo

Question:

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

42

77

35

82

Solution:

Coefficient of x10 in (x+x2+x3)7=coefficient of x3 in (1+x+x2)7=coefficient of x3 in (1−x3)7(1−x)-7
= 7+3-1C3 = 9C3 = 9×8×7/6 = 84
Alternate:
The digits are 1,1,1,1,1,2,3 or 1,1,1,1,2,2,2
Hence number of seven digit numbers formed = 7!/5! + 7!/(4!3!) = 42 + 35 = 77