Question

The value of the expression ${}^{47}{C}_4$ +${\sum }_{j=1}^5{}^{52-j}{C}_3$ is

• A. ${}^{51}{C}_4$
• B. ${}^{52}{C}_4$
• C. ${}^{52}{C}_3$
• D. ${}^{53}{C}_4$

Answer 1

Answer: (B)

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

${}^{47}{C}_4+{\sum }_{j=1}^5{}^{52-j}{C}_3=?{}^{}{}^{47}{C}_4{+}^{51}{C}_3{+}^{50}{C}_3{+}^{49}{C}_3{+}^{48}{C}_3{+}^{47}{C}_3$

We Know ,

${}^n{C}_r{+}^n{C}_{n-r}{=}^{n+1}{C}_r{\Rightarrow }^{47}{C}_4{+}^{47}{C}_3{+}^{48}{C}_3{+}^{49}{C}_3{+}^{50}{C}_3{+}^{51}{C}_3{\Rightarrow }^{48}{C}_4{+}^{48}{C}_3{+}^{49}{C}_3{+}^{50}{C}_3{+}^{51}{C}_3{\Rightarrow }^{49}{C}_4{+}^{49}{C}_3{+}^{50}{C}_3{+}^{51}{C}_3{\Rightarrow }^{50}{C}_4{+}^{50}{C}_3{+}^{51}{C}_3{\Rightarrow }^{51}{C}_4{+}^{51}{C}_3{\Rightarrow }^{52}{C}_4$

Hence the correct answer is ${}^{52}{C}_4$