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  • 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, isOption: 1 55<di

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

Option: 1

55


Option: 2

66


Option: 3

77


Option: 4

88


Answers (1)

best_answer

Let \mathrm{x_{i}(1 \leq i \leq 7)} be the digit at the \mathrm{i} th place. As \mathrm{1\leq x,\leq 3,and\: x_{1}+x_{2}+....+x_{3}= 10}. at most one \mathrm{x_{i}} can be 3.

Two cases arise.
Case 1.
Exactly one of \mathrm{x_{i}} is 3. In this case exactly one of the remaining \mathrm{x_{i}} is 2.
In this case, the number of seven digit numbers is
\frac{7 !}{5 !}=7 \times 6=42

Case 2.
None of x_{i} is 3
In this case exactly three of x_{i} is 2 and the remaining four x_{i} are 1.
In this case, the number of seven digit numbers is
\frac{7 !}{3 ! 4 !}=35

Hence, the required seven digit numbers is 42+35=77.

Posted by

Irshad Anwar

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