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6.Permutation and Combination
hard
$^n{C_r}{ + ^{n - 1}}{C_r} + ......{ + ^r}{C_r}$ =
A
$^{n + 1}{C_r}$
B
$^{n + 1}{C_{r + 1}}$
C
$^{n + 2}{C_r}$
D
${2^n}$
Solution
(b) $^r{C_r}{ + ^{r + 1}}{C_r}{ + ^{r + 2}}{C_r}……{ + ^{n – 1}}{C_r}{ + ^n}{C_r}$
${ = ^{r + 1}}{C_{r + 1}}{ + ^{r + 1}}{C_r}{ + ^{r + 2}}{C_r} + …..{ + ^{n – 1}}{C_r}{ + ^n}{C_r}$
${ = ^{r + 2}}{C_{r + 1}}{ + ^{r + 2}}{C_r} + …..{ + ^{n – 1}}{C_r}{ + ^n}{C_r}$
${ = ^{r + 3}}{C_{r + 1}} + ……{ + ^{n – 1}}{C_r}{ + ^n}{C_r}$.
On solving similar way, we get
$^{n – 1}{C_{r + 1}}{ + ^n}{C_r}{ + ^n}{C_r}{ = ^n}{C_{r + 1}}{ + ^n}{C_r}{ = ^{n + 1}}{C_{r + 1}}$.
Standard 11
Mathematics
