Permutations and Combinations 2 Question 13
13. The value of the expression ${ }^{47} C _4+\sum _{j=1}^{5}{ }^{52-j} C _3$ is
(a) ${ }^{47} C _5$
(b) ${ }^{52} C _5$
(c) ${ }^{52} C _4$
(d) None of these
(1980, 2M)
Match Type Question
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Answer:
Correct Answer: 13. (c)
Solution:
- Here, ${ }^{47} C _4+\sum _{j=1}^{5}{ }^{52-j} C _3$
$$ ={ }^{47} C _4+{ }^{51} C _3+{ }^{50} C _3+{ }^{49} C _3+{ }^{48} C _3+{ }^{47} C _3 $$
$$ =\left({ }^{47} C _4+{ }^{47} C _3\right)+{ }^{48} C _3+{ }^{49} C _3+{ }^{50} C _3+{ }^{51} C _3 $$
[using ${ }^{n} C _r+{ }^{n} C _{r-1}={ }^{n+1} C _r$ ]
$=\left({ }^{48} C _4+{ }^{48} C _3\right)+{ }^{49} C _3+{ }^{50} C _3+{ }^{51} C _3$
$=\left({ }^{49} C _4+{ }^{49} C _3\right)+{ }^{50} C _3+{ }^{51} C _3$
$=\left({ }^{50} C _4+{ }^{50} C _3\right)+{ }^{51} C _3$
$={ }^{51} C _4+{ }^{51} C _3={ }^{52} C _4$
