SATHEE: Permutations And Combinations Question 120

Permutations And Combinations Question 120

Question: $ ^{n}C _{r}{{+}^{n-1}}C _{r}+……{{+}^{r}}C _{r} $ = [AMU 2002]

Options:

A) $ ^{n+1}C _{r} $

B) $ ^{n+1}{C _{r+1}} $

C) $ ^{n+2}C _{r} $

D) $ 2^{n} $

Show Answer

Answer:

Correct Answer: B

Solution:

$ ^{r}C _{r}{{+}^{r+1}}C _{r}{{+}^{r+2}}C _{r}……{{+}^{n-1}}C _{r}{{+}^{n}}C _{r} $ $ {{=}^{r+1}}{C _{r+1}}{{+}^{r+1}}C _{r}{{+}^{r+2}}C _{r}+…..{{+}^{n-1}}C _{r}{{+}^{n}}C _{r} $ $ {{=}^{r+2}}{C _{r+1}}{{+}^{r+2}}C _{r}+…..{{+}^{n-1}}C _{r}{{+}^{n}}C _{r} $ $ {{=}^{r+3}}{C _{r+1}}+……{{+}^{n-1}}C _{r}{{+}^{n}}C _{r} $ . On solving similar way, we get $ ^{n-1}{C _{r+1}}{{+}^{n}}C _{r}{{+}^{n}}C _{r}{{=}^{n}}{C _{r+1}}{{+}^{n}}C _{r}{{=}^{n+1}}{C _{r+1}} $ .

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Permutations And Combinations Question 120

Question: $ ^{n}C _{r}{{+}^{n-1}}C _{r}+……{{+}^{r}}C _{r} $ = [AMU 2002]

Options:

Answer:

Solution:

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