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16 Sep 2014, 01:25
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In how many ways can we select 8 shirts from a closet containing 7 distinct red shirts and 5 distinct blue shirts, such that at least one shirt of each color remains in the closet, if the order of selection does not matter?
A. 460
B. 490
C. 493
D. 455
E. 445
A. 460
B. 490
C. 493
D. 455
E. 445
16 Sep 2014, 01:25
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Official Solution:
In how many ways can we select 8 shirts from a closet containing 7 distinct red shirts and 5 distinct blue shirts, such that at least one shirt of each color remains in the closet, if the order of selection does not matter?
A. 460
B. 490
C. 493
D. 455
E. 445
The total number of ways to select 8 shirts out of 12 distinct shirts (7 red and 5 blue) is given by the combination formula: \(C^8_{12}\).
The number of ways to select 8 shirts such that no red shirts are left in the closet is given by the product of the combinations of choosing all 7 red shirts and 1 blue shirts: \(C^7_7 * C^1_5\).
The number of ways to select 8 shirts such that no blue shirts are left in the closet is given by the product of the combinations of choosing all 5 blue shirts and 3 red shirts: \(C^5_5 * C^3_7\).
Therefore, the number of ways to select 8 shirts so that at least one red shirt and at least one blue shirt remain in the closet can be found by subtracting the sum of the two previous cases from the total number of ways to select 8 shirts: \(C^8_{12} - (C^7_7 * C^1_5 + C^5_5 * C^3_7) = 495 - (5 + 35) = 455\).
Answer: D
In how many ways can we select 8 shirts from a closet containing 7 distinct red shirts and 5 distinct blue shirts, such that at least one shirt of each color remains in the closet, if the order of selection does not matter?
A. 460
B. 490
C. 493
D. 455
E. 445
The total number of ways to select 8 shirts out of 12 distinct shirts (7 red and 5 blue) is given by the combination formula: \(C^8_{12}\).
The number of ways to select 8 shirts such that no red shirts are left in the closet is given by the product of the combinations of choosing all 7 red shirts and 1 blue shirts: \(C^7_7 * C^1_5\).
The number of ways to select 8 shirts such that no blue shirts are left in the closet is given by the product of the combinations of choosing all 5 blue shirts and 3 red shirts: \(C^5_5 * C^3_7\).
Therefore, the number of ways to select 8 shirts so that at least one red shirt and at least one blue shirt remain in the closet can be found by subtracting the sum of the two previous cases from the total number of ways to select 8 shirts: \(C^8_{12} - (C^7_7 * C^1_5 + C^5_5 * C^3_7) = 495 - (5 + 35) = 455\).
Answer: D
17 Sep 2023, 04:06
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Bunuel
Official Solution:
In how many ways can we select 8 shirts from a closet containing 7 distinct red shirts and 5 distinct blue shirts, such that at least one shirt of each color remains in the closet, if the order of selection does not matter?
A. 460
B. 490
C. 493
D. 455
E. 445
The total number of ways to select 8 shirts out of 12 distinct shirts (7 red and 5 blue) is given by the combination formula: \(C^8_{12}\).
The number of ways to select 8 shirts such that no red shirts are left in the closet is given by the product of the combinations of choosing all 7 red shirts and 1 blue shirts: \(C^7_7 * C^1_5\).
The number of ways to select 8 shirts such that no blue shirts are left in the closet is given by the product of the combinations of choosing all 5 blue shirts and 3 red shirts: \(C^5_5 * C^3_7\).
Therefore, the number of ways to select 8 shirts so that at least one red shirt and at least one blue shirt remain in the closet can be found by subtracting the sum of the two previous cases from the total number of ways to select 8 shirts: \(C^8_{12} - (C^7_7 * C^1_5 + C^5_5 * C^3_7) = 495 - (5 + 35) = 455\).
Answer: D
In how many ways can we select 8 shirts from a closet containing 7 distinct red shirts and 5 distinct blue shirts, such that at least one shirt of each color remains in the closet, if the order of selection does not matter?
A. 460
B. 490
C. 493
D. 455
E. 445
The total number of ways to select 8 shirts out of 12 distinct shirts (7 red and 5 blue) is given by the combination formula: \(C^8_{12}\).
The number of ways to select 8 shirts such that no red shirts are left in the closet is given by the product of the combinations of choosing all 7 red shirts and 1 blue shirts: \(C^7_7 * C^1_5\).
The number of ways to select 8 shirts such that no blue shirts are left in the closet is given by the product of the combinations of choosing all 5 blue shirts and 3 red shirts: \(C^5_5 * C^3_7\).
Therefore, the number of ways to select 8 shirts so that at least one red shirt and at least one blue shirt remain in the closet can be found by subtracting the sum of the two previous cases from the total number of ways to select 8 shirts: \(C^8_{12} - (C^7_7 * C^1_5 + C^5_5 * C^3_7) = 495 - (5 + 35) = 455\).
Answer: D
Hi Bunuel,
I used the following approach, is it correct? I do realize that it is a longer way.
Total shirts left behind = 12-8 =4
4 left behind can have 1 red and 3 blue
Selected shirts = 7C6 × 5C2 =7 × 10 = 70
4 left behind can have 2 red and 2 blue
Selected shirts = 7C5 × 5C3 = 21 × 10 = 210
4 left behind can have 3 red and 1 blue
Selected shirts = 7C4 × 5C4 = 35 x 5 = 175
Total combinations = 70 + 210 + 175 = 455
