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A box contains 5 different red and 6 different white balls. In how many ways can 6 balls be selected so that there are at least two balls of each colour
- a)425
- b)426
- c)452
- d)526
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
A box contains 5 different red and 6 different white balls. In how man...
Required number of ways = 6C2 × 5C4 + 6C3 × 5C3 + 6C4 × 5C2 = 75 + 200 + 150 = 425
Most Upvoted Answer
A box contains 5 different red and 6 different white balls. In how man...
Understanding the Problem
We need to select 6 balls from a box containing 5 different red and 6 different white balls, ensuring that there are at least 2 balls of each color.
Color Combinations
To satisfy the condition of at least 2 balls of each color, the valid combinations of red (R) and white (W) balls can be:
- 2 red and 4 white (2R, 4W)
- 3 red and 3 white (3R, 3W)
- 4 red and 2 white (4R, 2W)
Calculating Each Case
1. Case 1: 2 Red and 4 White
- Choose 2 red from 5: C(5, 2)
- Choose 4 white from 6: C(6, 4)
- Total ways = C(5, 2) * C(6, 4) = 10 * 15 = 150
2. Case 2: 3 Red and 3 White
- Choose 3 red from 5: C(5, 3)
- Choose 3 white from 6: C(6, 3)
- Total ways = C(5, 3) * C(6, 3) = 10 * 20 = 200
3. Case 3: 4 Red and 2 White
- Choose 4 red from 5: C(5, 4)
- Choose 2 white from 6: C(6, 2)
- Total ways = C(5, 4) * C(6, 2) = 5 * 15 = 75
Final Calculation
Now, sum the total ways from all cases:
- Total = 150 (2R, 4W) + 200 (3R, 3W) + 75 (4R, 2W)
- Total = 425
Thus, the total number of ways to select the balls is 425, corresponding to option 'A'.
We need to select 6 balls from a box containing 5 different red and 6 different white balls, ensuring that there are at least 2 balls of each color.
Color Combinations
To satisfy the condition of at least 2 balls of each color, the valid combinations of red (R) and white (W) balls can be:
- 2 red and 4 white (2R, 4W)
- 3 red and 3 white (3R, 3W)
- 4 red and 2 white (4R, 2W)
Calculating Each Case
1. Case 1: 2 Red and 4 White
- Choose 2 red from 5: C(5, 2)
- Choose 4 white from 6: C(6, 4)
- Total ways = C(5, 2) * C(6, 4) = 10 * 15 = 150
2. Case 2: 3 Red and 3 White
- Choose 3 red from 5: C(5, 3)
- Choose 3 white from 6: C(6, 3)
- Total ways = C(5, 3) * C(6, 3) = 10 * 20 = 200
3. Case 3: 4 Red and 2 White
- Choose 4 red from 5: C(5, 4)
- Choose 2 white from 6: C(6, 2)
- Total ways = C(5, 4) * C(6, 2) = 5 * 15 = 75
Final Calculation
Now, sum the total ways from all cases:
- Total = 150 (2R, 4W) + 200 (3R, 3W) + 75 (4R, 2W)
- Total = 425
Thus, the total number of ways to select the balls is 425, corresponding to option 'A'.
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A box contains 5 different red and 6 different white balls. In how many ways can 6 balls be selected so that there are at least two balls of each coloura)425b)426c)452d)526Correct answer is option 'A'. Can you explain this answer?
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