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In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?
- a)159
- b)194
- c)205
- d)209
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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In a group of 6 boys and 4 girls, four children are to be selected. In...
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In a group of 6 boys and 4 girls, four children are to be selected. In...
Since there should be at least one boy in the selection, the ways by which 4 children can be selected become 6C1*4C3, 6C2*4C2, 6C3*4C1, 6C4*4C0.
total selection becomes the sum of the above four combinations. ie, 24+90+80+15=209
total selection becomes the sum of the above four combinations. ie, 24+90+80+15=209
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In a group of 6 boys and 4 girls, four children are to be selected. In...
To solve this problem, we need to consider the different scenarios in which at least one boy is selected.
Scenario 1: Selecting 1 boy and 3 girls
In this scenario, we select 1 boy from the 6 available boys and 3 girls from the 4 available girls. The number of ways to do this is given by the combination formula:
C(6, 1) * C(4, 3) = 6 * 4 = 24
Scenario 2: Selecting 2 boys and 2 girls
In this scenario, we select 2 boys from the 6 available boys and 2 girls from the 4 available girls. The number of ways to do this is given by the combination formula:
C(6, 2) * C(4, 2) = 15 * 6 = 90
Scenario 3: Selecting 3 boys and 1 girl
In this scenario, we select 3 boys from the 6 available boys and 1 girl from the 4 available girls. The number of ways to do this is given by the combination formula:
C(6, 3) * C(4, 1) = 20 * 4 = 80
Scenario 4: Selecting 4 boys
In this scenario, we select 4 boys from the 6 available boys. The number of ways to do this is given by the combination formula:
C(6, 4) = 15
Total number of ways to select children with at least one boy:
24 + 90 + 80 + 15 = 209
Therefore, the correct answer is option D) 209.
Scenario 1: Selecting 1 boy and 3 girls
In this scenario, we select 1 boy from the 6 available boys and 3 girls from the 4 available girls. The number of ways to do this is given by the combination formula:
C(6, 1) * C(4, 3) = 6 * 4 = 24
Scenario 2: Selecting 2 boys and 2 girls
In this scenario, we select 2 boys from the 6 available boys and 2 girls from the 4 available girls. The number of ways to do this is given by the combination formula:
C(6, 2) * C(4, 2) = 15 * 6 = 90
Scenario 3: Selecting 3 boys and 1 girl
In this scenario, we select 3 boys from the 6 available boys and 1 girl from the 4 available girls. The number of ways to do this is given by the combination formula:
C(6, 3) * C(4, 1) = 20 * 4 = 80
Scenario 4: Selecting 4 boys
In this scenario, we select 4 boys from the 6 available boys. The number of ways to do this is given by the combination formula:
C(6, 4) = 15
Total number of ways to select children with at least one boy:
24 + 90 + 80 + 15 = 209
Therefore, the correct answer is option D) 209.
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