JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Permutations and Combinations

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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, is

(A)55
(B)66
(C)77✓ Correct
(D)88

Answer: (C)

Solution

We need seven-digit numbers using only digits 1, 2, and 3, with digit sum = 10.

Let the number have

a

ones,

b

twos, and

c

threes. Then:

$a + b + c = 7$ (seven digits)
$a + 2 b + 3 c = 10$ (sum = 10)

From these:

b + 2 c = 3

, giving solutions:

$c = 0, b = 3, a = 4$ : digits are 1,1,1,1,2,2,2 → $\frac{7 !}{4 ! 3 !} = 35$ arrangements
$c = 1, b = 1, a = 5$ : digits are 1,1,1,1,1,2,3 → $\frac{7 !}{5 !} = 42$ arrangements

Total =

35 + 42 = 77

JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Single correct option

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