Question

What is ${}^{47}{C}_4{+}^{51}{C}_3$ + ${\sum }_{j=2}^5$ ${}^{52-j}{C}_3$ equal to?

• A. ${}^{52}{C}_4$
• B. ${}^{51}{C}_5$
• C. ${}^{53}{C}_4$
• D. ${}^{52}{C}_5$

Answer 1

The correct option is A ${}^{52}{C}_4$

${}^{47}{C}_4{+}^{51}{C}_3+{\sum }_{j=2}^5{}^{52-j}{C}_3$

${=}^{47}{C}_4{+}^{51}{C}_3{+}^{50}{C}_3{+}^{49}{C}_3{+}^{48}{C}_3{+}^{47}{C}_3$

${=}^{51}{C}_3{+}^{50}{C}_3{+}^{49}{C}_3{+}^{48}{C}_3+{(}^{47}{C}_3{+}^{47}{C}_4)$

${=}^{51}{C}_3{+}^{50}{C}_3{+}^{49}{C}_3{+}^{48}{C}_3{+}^{48}{C}_4$ ...(${}^n{C}_r{+}^n{C}_{r+1}{=}^{n+1}{C}_{r+1}$)

${=}^{51}{C}_3{+}^{50}{C}_3{+}^{49}{C}_3{+}^{49}{C}_4$

${=}^{51}{C}_3{+}^{50}{C}_3{+}^{50}{C}_4$

${=}^{51}{C}_3{+}^{51}{C}_4$

${=}^{52}{C}_4$

So, the answer is option (A)