A box contains 5 red balls and 6 black balls. in how many ways can 6 balls be selected so that there are at least two balls of each color?
please give answer
i) 2 red,4 black => C(5,2)*C(6,4) =150
ii)3 red,3 black => C(5,3)*C(6,3) =200
iii)4red,2 black => C(5,4)*C(6,2) =75
Therefore total number of ways =
150+200+75 = 425 ways
Hope this helps! Cheers! :)
To find the number of ways to select 6 balls such that there are at least two balls of each color, we can consider different cases:
Case 1: Selecting exactly 2 red balls and 4 black balls.
- There are 5 red balls to choose from, and we need to select 2.
- There are 6 black balls to choose from, and we need to select 4.
- The number of ways to select 2 red balls from 5 is given by the binomial coefficient C(5, 2).
- The number of ways to select 4 black balls from 6 is given by the binomial coefficient C(6, 4).
- Therefore, the number of ways to select exactly 2 red balls and 4 black balls is C(5, 2) * C(6, 4).
Case 2: Selecting exactly 3 red balls and 3 black balls.
- There are 5 red balls to choose from, and we need to select 3.
- There are 6 black balls to choose from, and we need to select 3.
- The number of ways to select 3 red balls from 5 is given by the binomial coefficient C(5, 3).
- The number of ways to select 3 black balls from 6 is given by the binomial coefficient C(6, 3).
- Therefore, the number of ways to select exactly 3 red balls and 3 black balls is C(5, 3) * C(6, 3).
Case 3: Selecting 4 red balls and 2 black balls.
- There are 5 red balls to choose from, and we need to select 4.
- There are 6 black balls to choose from, and we need to select 2.
- The number of ways to select 4 red balls from 5 is given by the binomial coefficient C(5, 4).
- The number of ways to select 2 black balls from 6 is given by the binomial coefficient C(6, 2).
- Therefore, the number of ways to select 4 red balls and 2 black balls is C(5, 4) * C(6, 2).
Case 4: Selecting 5 red balls and 1 black ball.
- There are 5 red balls to choose from, and we need to select 5.
- There are 6 black balls to choose from, and we need to select 1.
- The number of ways to select 5 red balls from 5 is given by the binomial coefficient C(5, 5).
- The number of ways to select 1 black ball from 6 is given by the binomial coefficient C(6, 1).
- Therefore, the number of ways to select 5 red balls and 1 black ball is C(5, 5) * C(6, 1).
Now, to find the total number of ways to select 6 balls such that there are at least two balls of each color, we sum up the number of ways from each case:
Total number of ways = C(5, 2) * C(6, 4) + C(5, 3) * C(6, 3) + C(5, 4) * C(6, 2) + C(5, 5) * C(6, 1)