Binomial TheoremHard
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
Options
A.55
B.66
C.77
D.88
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 - 7
= 9C3
=
- 7 = 77.
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
.
coefficient of x3 in (1 + x + x2)7
coefficient of x3 in (1 - x3)7 (1 - x)-7
= 7+3-1C3 - 7
= 9C3
=
- 7 = 77.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
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