Permutations & Combinations
7-digit Numbers with Digits from {1,2,3,4} Summing to 12
nta_pyq_2023_apr
Grade 11
Question:
Number of 7-digit positive integers using only $\{1,2,3,4\}$ with digit sum $=12$ is_______.
Step-by-Step Solution
Key Concept: Find non-negative integer solutions of $d_1+d_2+\cdots+d_7=12$ where each $d_i\in\{1,2,3,4\}$. Substitute $e_i=d_i-1$: $\sum e_i=5$ with $e_i\in\{0,1,2,3\}$. Count arrangements.
413 numbers.
Correct Answer: 413