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  4. The value of 50C4+ displaystyle ∑r=16 56-rC3 is :

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Q. The value of $^{50}C_{4}+$ $\displaystyle \sum_{r=1}^6 $ $^{56-r}C_{3}$ is :

AIEEEAIEEE 2005Permutations and Combinations

Solution:

Key Idea : $^{n}C_{r}+^{n}C_{r+1}=^{n+1}C_{r+1}.$
Now, $^{50}C_{4}+^{55}C_{3}+^{54}C_{3}+^{53}C_{3}+^{52}C_{3}+^{51}C_{3}+^{50}C_{3}$
$=^{50}C_{3}+^{50}C_{4}+^{51}C_{3}+^{52}C_{3}+^{53}C_{3}+^{54}C_{3}+^{55}C_{3}$
$=^{51}C_{4}+^{51}C_{3}+^{52}C_{3}+^{53}C_{3}+^{54}C_{3}+^{55}C_{3}$
$\left(\because^{n}C_{r}+^{n}C_{r-1} =^{n+1}C_{r}\right)$
$=^{52}C_{4}+^{52}C_{3}+^{53}C_{3}+^{54}C_{3}+^{55}C_{3}$
$= ^{53}C_{4}+^{53}C_{3}+^{54}C_{3}+^{55}C_{3}$
$=^{54}C_{4}+^{54}C_{3}+^{55}C_{3}$
$=^{55}C_{4}+^{55}C_{3}=^{56}C_{4}$