Permutations & Combinations
Permutation Combination
nta_pyq_2025_jan
Grade 11
Question:
The number of different $5$-digit numbers greater than $50000$ that can be formed using the digits $0,1,2,3,4,5,6,7$, such that the sum of their first and last digits should not be more than $8$, is:
Step-by-Step Solution
Key Concept: Fix the first digit $d_{1}\in\{5,6,7\}$. For each, enumerate allowed $d_{5}$ with $d_{1}+d_{5}\le 8$. Middle three digits are free over the $8$-digit pool, contributing $8^{3}=512$. Subtract $50000$ itself (which is included but does not satisfy ``greater than'').
Valid $(d_{1},d_{5})$ pairs with $d_{1}\in\{5,6,7\}$ and $d_{1}+d_{5}\le 8$ ($d_{5}\in\{0,\dots,7\}$):
$d_{1}=5$: $d_{5}\in\{0,1,2,3\}$ — $4$ pairs.
$d_{1}=6$: $d_{5}\in\{0,1,2\}$ — $3$ pairs.
$d_{1}=7$: $d_{5}\in\{0,1\}$ — $2$ pairs.
Total pairs $=9$. Middle three digits: $8\cdot 8\cdot 8=512.$
Total counted $=9\cdot 512=4608.$ Subtract $50000$ (which is in the count but not $>50000$): $4608-1=4607.$
Correct Answer: 4