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  4. 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

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Q. 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

IIT JEEIIT JEE 2009Permutations and Combinations

Solution:

There are two possible cases
Case I Five $1 's$ , one $2's$, one $3's$
Number of numbers = $\frac{7!}{5!}=42$
Case II Four $1's$, three $2's$
Number of numbers =$\frac{7!}{4!3!}=35$
$\therefore $ Total number of numbers $= 42 + 35 = 77$