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Question: In a box, there are \( 10 \) balls, \( 4 \) are red, \( 3 \) black, \( 2 \) white and \( 1 \) yellow...

In a box, there are 1010 balls, 44 are red, 33 black, 22 white and 11 yellow. In how many ways can a child select 44 balls out of these 1010 balls? (Assume that the balls of the same colour are identical).
(A) 1010
(B) 1515
(C) 2020
(D) 2525

Explanation

Solution

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

Complete step-by-step answer :
If we have nn balls.
Then minimum number of balls
We can select is 00
And the maximum number of balls we can select is n.n.
To find the number of ways to select the ball from 00 to nn numbers
We can use the formula
Number of ways to select 00 to nn number of balls =1+x+x2+x3+....+xn= 1 + x + {x^2} + {x^3} + .... + {x^n} …………. (1)
Where, degree of xx denotes the number of balls that we have to select.
Now, consider the question.
We have 11 yellow ball.
So, using equation (1)
The number of ways to select 00 and 11 yellow ball
=1+x= 1 + x ………… (2)
We have 22 white balls
So, using equation (1)
The number of ways to select 00 to 22 white balls
=1+x+x2= 1 + x + {x^2} ……….. (3)
We have 33 black balls
So, using equation (1)
The number of ways to select 00 to 33 black balls
=1+x+x2+x3= 1 + x + {x^2} + {x^3} ……… (4)
We have 44 red balls
So, using equation (1)
The number of ways to select 00 to 44 red balls
=1+x+x2+x3+x4= 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+x2)(1+x+x2+x3)(1+x+x2+x3+x4)= (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+x2+x+x2+x3)(1+x+x2+x3+x4+x+x2+x3+x4+x5+x2+x3+x4+x5+x6+x3+x4+x5+x6+x7)= (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+2x2+x3)(1+2x+3x2+4x3+4x4+3x5+2x6+x7)= (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 44 balls.
This is represented by the coefficient of x4.{x^4}.
Therefore, we should add the coefficients of x4.{x^4}.
This can be above by neglecting all other terms.
=(4+8+6+2)x4= (4 + 8 + 6 + 2){x^4}
=20x4= 20{x^4}
Therefore, there are 2020 ways to select 44 balls out of 1010 given balls.
Therefore, from the above explanation the correct option is (C) 20.20.
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 xx greater than 4.4. because, we only 00 needed to calculate the number of ways of selecting 44 balls.
This would have saved us from writing equation (4).