Left( {begin{array}{*{20}{c}}n{n - r}end{array}} right), + ,left( {begin{array}{*{20}{c}}n{r + 1}end

English

Gujarati

Hindi

6.Permutation and Combination

normal

$\left( {\begin{array}{{20}{c}}n\\{n - r}\end{array}} \right)\, + \,\left( {\begin{array}{{20}{c}}n\\{r + 1}\end{array}} \right)$, whenever $0 \le r \le n - 1$is equal to

$\left( {\begin{array}{*{20}{c}}n\\{r - 1}\end{array}} \right)$

$\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right)$

$\left( {\begin{array}{*{20}{c}}n\\{r + 1}\end{array}} \right)$

$\left( {\begin{array}{*{20}{c}}{n + 1}\\{r + 1}\end{array}} \right)$

Solution

(d)$\left( \begin{array}{l}\,\,\,n\,\\\,n – r\end{array} \right)$+$\left( \begin{array}{l}\,\,\,n\,\\r + 1\end{array} \right)$ = $^n{C_{n – r}}{ + ^n}{C_{r + 1}}$
$ \Rightarrow {\,^n}{C_r}\, + {\,^n}{C_{r + 1}}$ = $^{n + 1}{C_{r + 1}} = \left( {\begin{array}{*{20}{c}}{n + 1}\\{r + 1}\end{array}} \right)$.

Standard 11

Mathematics

In an election there are $5$ candidates and three vacancies. A voter can vote maximum to three candidates, then in how many ways can he vote

easy

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and $2$ different consonants can be formed from the alphabet?

medium

The number of words from the letters of the word $'RAJASTHAN' $ by taking all the letters at a time in which vowels are alternate, are

normal

If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^n{C_{r – 1}}$$ + {\,^n}{C_r} = $

normal

If $\sum\limits_{i = 0}^4 {^{4 + 1}} {C_i} + \sum\limits_{j = 6}^9 {^{3 + j}} {C_j} = {\,^x}{C_y}$ ($x$ is prime number), then which one of the following is incorrect

normal

$\left( {\begin{array}{*{20}{c}}n\\{n - r}\end{array}} \right)\, + \,\left( {\begin{array}{*{20}{c}}n\\{r + 1}\end{array}} \right)$, whenever $0 \le r \le n - 1$is equal to

Solution

Similar Questions

In an election there are $5$ candidates and three vacancies. A voter can vote maximum to three candidates, then in how many ways can he vote

The English alphabet has $5$ vowels and $21$ consonants. How many words with two different vowels and $2$ different consonants can be formed from the alphabet?

The number of words from the letters of the word $'RAJASTHAN' $ by taking all the letters at a time in which vowels are alternate, are

If $n$ and $r$ are two positive integers such that $n \ge r,$ then $^n{C_{r – 1}}$$ + {\,^n}{C_r} = $

If $\sum\limits_{i = 0}^4 {^{4 + 1}} {C_i} + \sum\limits_{j = 6}^9 {^{3 + j}} {C_j} = {\,^x}{C_y}$ ($x$ is prime number), then which one of the following is incorrect

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$\left( {\begin{array}{{20}{c}}n\\{n - r}\end{array}} \right)\, + \,\left( {\begin{array}{{20}{c}}n\\{r + 1}\end{array}} \right)$, whenever $0 \le r \le n - 1$is equal to