Question

A computer programming team has 17 members. (For each answer, enter an exact number.) (a) How many ways can a group of nine be chosen to work on a project? 24310 Correct: Your answer is correct. (b) Suppose nine team members are women and eight are men. (i) How many groups of nine can be chosen that contain five women and four men? 1 Incorrect: Your answer is incorrect. (ii) How many groups of nine can be chosen that contain at least one man? 9 Incorrect: Your answer is incorrect. (iii) How many groups of nine can be chosen that contain at most three women? (c) Suppose two team members refuse to work together on projects. How many groups of nine can be chosen to work on a project? (d) Suppose two team members insist on either working together or not at all on projects. How many groups of nine can be chosen to work on a project?

Answer 1

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.

Reasons:

The combination of n objects taking r at a time is given as follows

$_nC_r = \dbinom{n}{r}$

(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 = $\dbinom{17}{9}$ = 24,310 ways.

(b) (i) Number of women = 9

Number of men = 8

Number of groups of 9 containing 5 women and 4 men = $\dbinom{9}{5} \times \dbinom{8}{4}$

$\dbinom{9}{5} \times \dbinom{8}{4} = 126 \times 70 = 8,820$

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 $\dbinom{17}{9} - \dbinom{9}{9}$

$\dbinom{17}{9} - \dbinom{9}{9} = 24,310 - 1 = 24,309$

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;

$\dbinom{8}{8} \times \dbinom{9}{1} + \dbinom{8}{7} \times \dbinom{9}{2} + \dbinom{8}{6} \times \dbinom{9}{3} = 2,649$

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 $\dbinom{15}{9} + 2 \times \dbinom{15}{8}$

$\dbinom{15}{9} + 2 \times \dbinom{15}{8} = 5,005 + 2 \times 6,435 = 17,875$

(d) Two team members must work together or not at all is $\dbinom{15}{7} - \dbinom{15}{9}$

$\dbinom{15}{7} - \dbinom{15}{9} = 6,435 - 5,005 = 1,430$

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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