Question

The number of ways of choosing (x+8) balls out of 36 balls is equal to choosing x balls out of 36 balls. Find the number of ways of choosing (x+5) balls out of 25 balls. ​

Answer 1

$\large\underline{\sf{Solution-}}$

We know,

Number of ways in which r objects can be selected from n distinct objects is

$\boxed{ \bf{ \:^nC_r \: \: = \: \: \frac{n!}{ r! \: (n - r)!}}}$

So,

The number of ways of choosing (x+8) balls out of 36 balls is given by

$\rm \: = \: ^{36}C_{x + 8}$

And

The number of ways of choosing x balls out of 36 balls is given by

$\rm \: = \: ^{36}C_{x}$

According to statement,

The number of ways of choosing (x+8) balls out of 36 balls is equals to the number of ways of choosing x balls out of 36 balls.

$\bf\implies \:^{36}C_{x + 8} = \: ^{36}C_{x}$

Now, we know,

$\boxed{ \bf{ \:^nC_x \: = ^nC_y \: \rm \implies\:x = y \: \: or \: \: n = x + y}}$

So, using this identity,

$\rm :\longmapsto\:x + 8 \: \ne \: x$

Hence,

$\rm :\longmapsto\:36 = x + x + 8$

$\rm :\longmapsto\:36 = 2x + 8$

$\rm :\longmapsto\:36 - 8 = 2x$

$\rm :\longmapsto\:28 = 2x$

$\bf\implies \:x = 14$

So,

The number of ways of choosing (x+5) balls out of 25 balls

is given by

$\rm \: = \: ^{25}C_{14 + 5}$

$\rm \: = \: ^{25}C_{19}$

$\rm \: = \: \dfrac{25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19!}{19!}$

$\rm \: = \: 25 \times 24 \times 23 \times 22 \times 21 \times 20 \times 19$

$\rm \: = \: 127512000$

Additional Information :-

$\boxed{ \bf{ \:^nC_r + \: ^nC_{r - 1} = \: \: ^{n + 1}C_{r}}}$

$\boxed{ \bf{ \:^nC_r = \: \dfrac{n}{r} \: ^{n - 1} \: C_{r - 1}}}$

Answer 2

The answer is 127512000 ways

Formula: nCr = n!/[r! (n-r)!]

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