Permutations & Combinations
Double sum with min/max of binomials
MJMT_Full_Test_07
Grade 12
Question:
Let $A=\sum_{i=0}^{2023}\sum_{j=0}^{2023}\min\{\binom{2023}{i},\binom{2023}{j}\}$ and $B=\sum_{i=0}^{2023}\sum_{j=0}^{2023}\max\{\binom{2023}{i},\binom{2023}{j}\}$. Then $A+B$ equals
$2024\cdot2^{2023}$
$2023\cdot2^{2023}$
$2024(2^{2023}-1)$
$2023(2^{2023}-1)$
Step-by-Step Solution
Key Concept: $\min(a,b)+\max(a,b)=a+b$. So $A+B=\sum_{i,j}(\binom{n}{i}+\binom{n}{j})=2\cdot2024\sum_i\binom{n}{i}=2\cdot2024\cdot2^{2023}$? No: $A+B=\sum_{i,j}(C_i+C_j)=2024\sum_i C_i\cdot2024... $
$A+B=2024\cdot2^{2023}$.
Correct Answer: 1