An engineering company has four openings and the applicant pool consists of 6 database administrators and eight network engineers. All are equally qualified so the hiring will be done randomly. The hiring committee consists of 4 women and 4 men. a. If one person on the hiring committee is chosen at random to draw the names out of a hat, what is the probability that the person drawing the names is a woman. b. How many ways can the group who are to be hired be formed if they are no restrictions on composition? c. How many ways can 3 database administrators be chosen?

Question

An engineering company has four openings and the applicant pool consists of 6 database administrators and eight network engineers. All are equally qualified so the hiring will be done randomly. The hiring committee consists of 4 women and 4 men.  a. If one person on the hiring committee is chosen at random to draw the names out of a hat, what is the probability that the person drawing the names is a woman.  b. How many ways can the group who are to be hired be formed if they are no restrictions on composition?  c. How many ways can 3 database administrators be chosen?

May Dunn · Accepted Answer

Step 1  Introduction:   According to the combination rule, if r distinct items are to be chosen from n items without replacement, in such a way that the order of selection does not matter, then the selection can be made in ({n}Cr)=n!r!(n–r)!=n(n–1)…(n–r+1)r! ways.  Step 2  a. Total number of people in the hiring committee is, 4 men+4 women=8.   Thus, without any constraint, the total number of ways in which r=1 member can be chosen from among the n=8 members of the hiring committee is ({8}C1).   ({8}C1)   =8!1!⋅(8–1)!   =8!1!∙7!   =8.   In the hiring committee, number of members who are women is 4.   Thus, the total number of ways in which r=1 woman member can be chosen from among the n=4 women members of the hiring committee is ({4}C1).   ({4}C1)   =4!1!⋅(3−1)!   =4!1!⋅3!   =4.   Thus, the probability that the person drawing names is a woman is, 48=12 or 0.5.  Step 4   b. Total number of members in the applicant pool is, 6 database administrators +8 ≠twork eng∈eers=14.   Out of these applicants, 4 can be chosen to fill up the 4 openings available.   Thus, without any constraint or restriction on the composition of the group to be hired, the total number of ways in which r=4 members can be chosen from among the n=14 members in the applicant pool is ({14}C4).   ({14}C4)   =14!4!⋅(14−4)!   =14!4!⋅10!   =14⋅13⋅12⋅114⋅3⋅2⋅1   =1001.   Thus, without any restriction on the composition of the group to be hired, the number of possible ways in which the group can be formed is 1,001.   Step 5   c. The applicant pool has 6 database administrators.   Out of these, 3 are to be chosen into the group to be hired.   Thus, the total number of ways in which r=3 database administrators can be chosen from among the n=6 database administrators in the application pool is ({6}C3).   ({6}C3)   =6!3!⋅(6−3)!   =6!3!⋅3!   =6⋅5⋅43⋅2⋅1   =20.   Thus, the number of possible ways in which 3 database administrators can be chosen from among the 6 database administrators in the application pool is 20.

An engineering company has four openings and the applicant pool consis

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