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STD 11 - 6. permutation and combination — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 6. permutation and combination4 Marks
MCQ
$^n{C_r}{ + ^{n - 1}}{C_r} + ......{ + ^r}{C_r}$ =
  • A
    $^{n + 1}{C_r}$
  • ✓
    $^{n + 1}{C_{r + 1}}$
  • C
    $^{n + 2}{C_r}$
  • D
    ${2^n}$
✓

Answer

Correct option: B.
$^{n + 1}{C_{r + 1}}$
b
(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

$^{n - 1}{C_{r + 1}}{ + ^n}{C_r}{ + ^n}{C_r}{ = ^n}{C_{r + 1}}{ + ^n}{C_r}{ = ^{n + 1}}{C_{r + 1}}$.

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