Discrete Math: 1) Your school\'s computer club has 25 members, of which 5 are fr

Question

Discrete Math:

1) Your school's computer club has 25 members, of which 5 are freshmen, 5 are sophmores, 10 are juniors and 5 are seniors

a) How many ways can a club president be selected if freshmen are ineligible to be president?

b) how many ways are there to select two seniors to serve on your school's activity council?

c) how many ways are there to choose a club president and vice president if the president and vice president cannot be from the same class and freshmen cannot hold either position?

d) how many ways are there to choose a club subcommittee with 25 members?

2) There are 10 people at a party. If every person shakes hands with everyone else at the party, then how many total handshakes take place?

3) A new car can be ordered with a choice of 10 exterior colors, 7 interior colors, with or without air conditioning, with or without power steering, and with or without a power package. assuming all possible combinations are allowable, how many possible versions of this car can one order?

Explanation / Answer

Preface

All the questions require knowledge of the theory of Combination. The concepts relevant the present questions are:

1. Any r things can be selected from n things in nCr ways.

2. nCr = (n!)/{(r!)(n - r)!}

3 For 'or' selections add the possibilities and 'and' selections multiply.

4. Good to remember the following simplifications:

nCn = nC0 = 1

nC1 = n

nCr = nCn - r   

0! = 1

n! = n(n - 1)(n - 2) ........ 1

n! can re written as n(n - 1)! or n(n - 1)(n - 2)! and so on.

Answer to Q1

Part (a)

Out of 25 club members, 5 are freshers who are not eligible to be selected as President. So, one member out of the remaining 20 members can be selected as President. Selecting any one out of 20 is given by20C2 = (201)/{(1!)(20 - 1)!} = {20(19!)}/{(1!)(19!)} = 20. ANSWER

Part (b)

There are 5 seniors, of which 2 are to be selected. This can be done in

5C2 = (51)/{(2!)(5 - 2)!} = {(5 x 4 x 3!)}/{(2!)(3!)} = (5 x 4)/2 = 10 ANSWER

Part (c)

Out of 25 members, 5 freshers are not eligible. Out of the remaining 20 members, President can be a sohpomore(1 out of 10) and Vice-president can be a non-sophomore (1 out of 10), or President can be a junior (1 out of 5) and Vice-president can be a non-junior (1 out of 15) or President can be a senior (1 out of 5)and Vice-president can be a non-senior (1 out of 15).

Number of possible selections will be:( 10C1 x 10C1) +  ( 5C1 x 15C1) + ( 5C1 x 15C1) = (10 x 10) + (5 x 15) + (5 x 15) = 100 + 2(75) = 100 + 150 = 250 - ANSWER , , [Note that nC1 = n]

Part (d)

There are 25 members and all of them must be selected since sub-committee also should have 25 members. This can be in 25C25 = 1. ANSWER [Note that nCn = 1]

Answer to Q 2

One hand-shake occurs between two persons. [Note that person A shaking hands with person B is the same as person B shaking hands with person A]. So, total number of hand-shakes will be 10C2 = (101)/{(2!)(10 - 2)!} = {10 x 9 x 8!)}/{(2!)(8!)} = (10 x 9)/2.= 45 ANSWER

Answer to Q 3

There are 10 options for Exterior colours of which only one is to be selcted. This can be done in 10C1 = (101)/{(1!)(10 - 1)!} = {10 x 9!)}/{(1)(9!)} = 10

Similarly one out of 7 Interior colours can be selected in 7C1 = (71)/{(1!)(7 - 1)!} = {7 x 6!)}/{(1)(6!)} = 7

For the remaining 3 features, each has 2 possibilities i.e., with or without.

Thus, total number of selections = 10 x 7 x 2 x 2 x 2 = 560 ANSWER

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