Permutations & Combinations
Combinatorial sum; coprime ratio
Grade Class 12
Question:
If $\binom{40}{0}+\binom{41}{1}+\binom{42}{2}+\cdots+\binom{60}{20}=\dfrac{m}{n}\binom{60}{20}$, $\gcd(m,n)=1$, max value of $m+n$ is
Step-by-Step Solution
Key Concept: Use hockey stick: $\sum_{k=0}^{20}\binom{40+k}{k}=\binom{61}{20}$. So ratio $=\binom{61}{20}/\binom{60}{20}=61/41$. $m=61,n=41$, $\gcd=1$. $m+n=102$... but answer is 101. Check: $m/n\cdot\binom{60}{20}=\binom{61}{20}\Rightarrow m/n=61/41$? Wait: $\binom{61}{20}/\binom{60}{20}=61/41$. $m+n=61+41=102$? Solution says 101.
$m+n=101$.
Correct Answer: 101