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The number of ways in which 4 balls can be selected from a bag containing 4 identical and 4 different balls is

(a) 120

(b) 80

(c) 60

(d) 16

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Answer 1

Answer:

D) 16

Step-by-step explanation:

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

Answer 2

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

Given that,

There are 4 identical balls and 4 different balls.

We have to find the number of ways in which 4 balls can be selected.

So,

Following cases arises :-

$\begin{gathered}\boxed{\begin{array}{c|c} \sf Number \: ofidentical \: balls & \sf Number \: of \: distinct \: balls \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 4 \\ \\ \sf 1 & \sf 3\\ \\ \sf 2 & \sf 2\\ \\ \sf 3 & \sf 1\\ \\ \sf 4 & \sf 0 \end{array}} \\ \end{gathered}$

Now,

Case - 1

Number of ways in which 0 identical ball and 4 distinct ball can be selected from 4 identical balls and 4 distinct balls is

$\rm \: = \:1 \times \: ^4C_{4}$

$\rm \: = \:1 \times 1$

$\rm \: = \:1$

Case - 2

Number of ways in which 1 identical ball and 3 distinct ball can be selected from 4 identical balls and 4 distinct balls is

$\rm \: = \:1 \times \: ^4C_{3}$

$\rm \: = \:1 \times 4$

$\rm \: = \:4$

Case - 3

Number of ways in which 2 identical ball and 2 distinct ball can be selected from 4 identical balls and 4 distinct balls is

$\rm \: = \:1 \times \: ^4C_{2}$

$\rm \: = \:\dfrac{4 \times 3}{2 \times 1}$

$\rm \: = \:6$

Case - 4

Number of ways in which 3 identical ball and 1 distinct ball can be selected from 4 identical balls and 4 distinct balls is

$\rm \: = \:1 \times \: ^4C_{1}$

$\rm \: = \:1 \times 4$

$\rm \: = \:4$

Case - 5

Number of ways in which 4 identical ball and 0 distinct ball can be selected from 4 identical balls and 4 distinct balls is

$\rm \: = \:1 \times \: ^4C_{0}$

$\rm \: = \:1 \times 1$

$\rm \: = \:1$

Hence,

Total number of ways in which 4 ball can be selected from 4 identical balls and 4 distinct balls is

$\rm \: = \:1 + 4 + 6 + 4 + 1$

$\rm \: = \:16$

Hence, Option (d) is correct.

Concept Used :-

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

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: ^nC_{r} = \frac{n!}{r! \: (n - r)!} \: \: }}}$

Also,

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: ^nC_{0} \: = \: ^nC_{n} \: = 1 \: \: }}}$

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: ^nC_{1} \: = \: ^nC_{n - 1} \: = n \: \: }}}$

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: ^nC_{2} \: = \: \frac{n(n - 1)}{2} \: }}}$