Permutations and Combinations
Properties of Combinations
IIT-JEE 1982
Grade 11
Question:
The value of $^{47}C_4 + \sum_{j=1}^5 {^{52-j}C_3}$ equals:
(1) $^{52}C_4$
(2) $^{52}C_5$
(3) $^{51}C_4$
(4) $^{53}C_4$
Step-by-Step Solution
Key Concept: Write the sum as $^{47}C_4+^{51}C_3+^{50}C_3+^{49}C_3+^{48}C_3+^{47}C_3$. Apply Pascal's identity $^nC_r+^nC_{r-1} = ^{n+1}C_r$ repeatedly: combine $^{47}C_4 + ^{47}C_3 = ^{48}C_4$, then keep combining upward to reach $^{52}C_4$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)