please solve all parts. Solve for total possible combinations. The department is

Question

please solve all parts. Solve for total possible combinations.

The department is forming a committee to work out the details of a new course. The committee will consist of a president, vice president, and secretary, along with 7 at-large members (meaning they hold no special position, so are interchangeable). (As usual in this chapter, your answers should not be simplified.) (a) The department consists of only 10 faculty. (b) The department consists of 21 faculty. (c) The department consists of 21 faculty, but 8 of those are non-tenure-track and can only serve as at-large members. (d) The department consists of 21 (full) faculty, but Vi and Fi cannot hold the president/vice president roles together (at most one of them can have such a role).

Explanation / Answer

(a) Since there are only 10 faculty, the president can be chosen in 10 ways, the vice-president in 9 (except the president) ways and the seceratary in 8 ways. The remaining 7 will be at-large members

So the total number of combinations is 10*9*8

(b) Like in the preceding problem, the president can be chosen in 21 ways, the vice-president in 20, the seceratary in 19 ways.

However, there are 18 more people for the 7 at-large positions. So the number of ways this can be done is 18C7

So the total no of combinations = 21*20*19*(18C7)

(c) Since 8 cannot serve as president, vice-president or seceratary, the number of ways the president can be chosen is

(21-8) ways = 13 ways.

Similarly, the vice-president and the seceratary can be chosen in 12 and 11 ways respectively.

Now the at-large members could be any of the 21-3 members (three positions have been occupied).

So the no of ways this can be done is 18C7

Thus the total no of combinations = 13 * 12 * 11 * 18C7

(d) We will break the problem down into three parts

(i) When neither Vi and Fi have any special post (president or vice-president)

So the role of president and vice-president can be chosen in 19 and 18 ways respectively.

The role of the seceratary can be chosen in 19 ways (two positions are filled)

The role of the at-large members are filled in the usual 18C7 ways

Total no of combinations = 19*18*18C7

(ii) One of Vi and Fi hold a special position (president or vice-president)

This can be done in four ways. (Vi as president, Vi as vice-president, Fi as president, Fi as vice-president)

Then the other special position can be chosen in 19 ways.

The seceratary can be chosen in 19 ways (two positions are filled)

The seven at-large positions can be filled in the usual 18C7 ways

So the no of combinations = 4 * 19 * 19 * 18C7

Therefore the total no combinations = 19*18*18C7 + 4 * 19 * 19 * 18C7

Here nCk denotes the number of combinations that is no ways in which we can choose k objects out of n objects. This is given by the formula nCk = n! / [ (n-k)! k! ]

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.