Permutations & Combinations
Counting S and P Sets — n(S)+n(P)
nta_pyq_2026_jan
Grade 11
Question:
Let $S$ denote the set of 4-digit numbers $abcd$ such that $a>b>c>d$ and $P$ denote the set of 5-digit numbers having product of its digits equal to 20. Then $n(S)+n(P)$ is equal to _____.
Step-by-Step Solution
Key Concept: $n(S)$: choose 4 distinct digits from $\{0,1,\ldots,9\}$ and arrange in decreasing order $=\binom{10}{4}=210$. $n(P)$: 5-digit numbers with digit product $20=2^2\times5$. All digits $\geq1$. Must include exactly one 5; remaining 4 digits multiply to 4.
$n(S)=210$, $n(P)=50$. Sum $=260$.
Correct Answer: 260