JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Permutations and Combinations

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _____.

Answer: 413

Solution

The count equals the number of solutions of

x_{1} + \dots + x_{7} = 12

with each

x_{i} \in {1, 2, 3, 4}

. Putting

y_{i} = x_{i} - 1 \in {0, 1, 2, 3}

turns this into

y_{1} + \dots + y_{7} = 5

with each

y_{i} \leq 3

. Without the upper bound the number of non-negative solutions is

(6 11) = 462

. The bound fails only when some

y_{i} \geq 4

; fixing such a variable leaves a sum of

1

to distribute, giving

(6 7) = 7

solutions, and with

7

choices of variable this removes

7 \times 7 = 49

. Two variables cannot both reach

4

since the total is only

5

, so the number of integers is

462 - 49 = 413

JEE Main · 2023 · 13 Apr 2023 · Shift 1 · Integer answer