Question

In a box, there are $ 10 $ balls, $ 4 $ are red, $ 3 $ black, $ 2 $ white and $ 1 $ yellow. In how many ways can a child select $ 4 $ balls out of these $ 10 $ balls? (Assume that the balls of the same colour are identical).
(A) \[10\]
(B) \[15\]
(C) \[20\]
(D) \[25\]

Answer 1

Hint: We can write a polynomial expression of degree to denote the possible ways of selecting any ball. Where the degree of $ x $ represents the number of balls drowned.

Complete step-by-step answer:
If we have $ n $ balls.
Then minimum number of balls
We can select is $ 0 $
And the maximum number of balls we can select is $ n. $
To find the number of ways to select the ball from $ 0 $ to $ n $ numbers
We can use the formula
Number of ways to select $ 0 $ to $ n $ number of balls $ = 1 + x + {x^2} + {x^3} + .... + {x^n} $ …………. (1)
Where, degree of $ x $ denotes the number of balls that we have to select.
Now, consider the question.
We have $ 1 $ yellow ball.
So, using equation (1)
The number of ways to select $ 0 $ and $ 1 $ yellow ball
$ = 1 + x $ ………… (2)
We have $ 2 $ white balls
So, using equation (1)
The number of ways to select $ 0 $ to $ 2 $ white balls
$ = 1 + x + {x^2} $ ……….. (3)
We have $ 3 $ black balls
So, using equation (1)
The number of ways to select $ 0 $ to $ 3 $ black balls
 $ = 1 + x + {x^2} + {x^3} $ ……… (4)
We have $ 4 $ red balls
So, using equation (1)
The number of ways to select $ 0 $ to $ 4 $ red balls
$ = 1 + x + {x^2} + {x^3} + {x^4} $ ……….. (5)
Total number of possible ways of selecting any number of balls is the product of all possible ways for selecting all the balls.
$ = (1 + x)(1 + x + {x^2})(1 + x + {x^2} + {x^3})(1 + x + {x^2} + {x^3} + {x^4}) $
Open first two and last two brackets
$ = (1 + x + {x^2} + x + {x^2} + {x^3})(1 + x + {x^2} + {x^3} + {x^4} + x + {x^2} + {x^3} + {x^4} +
{x^5} + {x^2} + {x^3} + {x^4} + {x^5} + {x^6} + {x^3} + {x^4} + {x^5} + {x^6} + {x^7}) $
$ = (1 + 2x + 2{x^2} + {x^3})(1 + 2x + 3{x^2} + 4{x^3} + 4{x^4} + 3{x^5} + 2{x^6} + {x^7})
$
Now, open these to brackets as well
We need a number of ways of selecting $ 4 $ balls.
This is represented by the coefficient of $ {x^4}. $
Therefore, we should add the coefficients of $ {x^4}. $
This can be above by neglecting all other terms.
 $ = (4 + 8 + 6 + 2){x^4} $
 $ = 20{x^4} $
Therefore, there are $ 20 $ ways to select $ 4 $ balls out of $ 10 $ given balls.
Therefore, from the above explanation the correct option is (C) $ 20. $
So, the correct answer is “Option C”.

Note: We could have avoided the lengthy calculation by negating all the terms that have a degree of $ x $ greater than $ 4. $ because, we only $ 0 $ needed to calculate the number of ways of selecting $ 4 $ balls.
This would have saved us from writing equation (4).