The number of groups that can be chosen from the team of 17 depends on the given conditions of the selection.
The correct responses are;
(a) The number of ways of choosing groups of 9 from the 17 members are 24,310 ways.
(b) (i) The number of groups of 9 containing 5 women and 4 men are 8,820 groups.
(ii) The number of groups of 9 that contain at least one man is 24,309 groups.
(iii) The number of groups of 9 that contain at most 3 women are 2,649 groups.
(c) If two team members refuse to work together on projects, the number of groups of 9 that can be chosen are 17,875 groups.
(d) If two team members must work together or not at all the number of groups of 9 that can be chosen are 1,430 groups.
Given,
The combination of n objects taking r at a time is given as follows
ₙCr = (ⁿr)
(a) The number of ways of choosing groups of 9 from the 17 members is given as follows;
Number of ways of choosing groups of 9 = (17, 9) = 24,310 ways.
(b) (i) Number of women = 9
Number of men = 8
Number of groups of 9 containing 5 women and 4 men = (9, 5) x (8, 4)
(9, 5) x (8, 4) = 126 x 70 = 8820
Number of groups of 9 containing 5 women and 4 men = 8,820 groups.
(ii) The number of groups that contain at least one man is (17, 9) - (9, 9)
(17, 9) - (9, 9) = 24,310 - 1 = 24,309
The number of groups that contain at least one man are 24,309 groups.
(iii) The number of groups of 9 that contain at most 3 women if given as follows;
(8, 8) × (9, 1) + (8, 7) × (9, 2) + (8, 6) × (9, 3) = 2649
The number of groups of 9 that contain at most 3 women are 2,649 groups.
(c) Given that two team members refuse to work together on projects, the number of groups of 9 that can be chosen are (15, 9) + 2 × (15, 8)
(15, 9) + 2 × (15, 8) = 5,005 + 2 × 6,435 = 17,875
(d) Two team members must work together or not at all is (15, 7) - (15, 9)
(15, 7) - (15, 9) = 6,435 - 5,005 = 1,430
The number of groups of 9 that can be chosen two team members must work together or not at all are 1,430 groups.
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