Answer to Question #190231 in Discrete Mathematics for Maricel

Question

. A special type of password consists of six(6) different letters of the alphabet, where each letter is used only once. How

many different possible passwords are there? (3 pts)

A. 244, 140, 625

B. 308, 915, 776

C. 165,765,600

D. 213,127,200 

A. COUNTING METHODS (5 pts each)

1. How many strings of length 4 can be formed using the letters ABCDE if it starts with letters AC and repetition is not

allowed?

2. There are 10 multiple choice questions in an examination. Each of the questions have four choices. In how many ways

can an examinee give possible answers?

B. BINOMIAL COEFFICIENTS

Expand (2𝑥 + 4𝑎) 4 using the binomial theorem. (10 pts)

C. PIGEONHOLE PRINCIPLE (5 pts) . Explain briefly.

Do you agree that there are 3 persons who have the same first and last name? Why and why not? 

Answer 1

Solution.

English alphabet has 26 letters. From them the password from 6 different letters can be made in $26•25•24•23•22•21=165765600$different ways.

Answer. C.

A.

1. 3•2=6 ways.

2. We need the product rule:

4•4•4•4•4•4•4•4•4•4=$4^{10}=1048576$ ways.

B.

$(2a+4b)^4=16a^4+4•8a^3•4b+6•4a^2•16b^2+4•2a•64b^3+256b^4=16a^4+128a^3b+384a^2b^2+512ab^3+256b^4.$

C.

No.

The pigeonhole principle states that if n items are put into m containers, with n>m, then at least one container must contain more than one item. We have $n\in N$ people and $m=3$ various for their families. $n>3,$ there are many (at least four) people in the world who have the same first and last name families.