Question

1. A password is made out of 8 characters. The password must contain exactly 8 characters and can consist of any uppercase letter, lowercase letter, number, or pound character. How many possible passwords are there given the following specification:

a) password cannot have a number character.

b) password must have exactly 1-pound (#) character that is not at the beginning or end of a variable name.

c) password must have at least one number.

2. Alison has 10 bananas and 15 pineapples. He realized that he only can eat 6 fruits today. If she does not care about the order each fruit has to be eaten, how many ways are there to eat the fruits if:

1. she would like to eat at least as many pineapples as bananas?
2. She would like to eat as many bananas as pineapples, but she cannot eat the banana without eating the pineapple.

Answer 1

In the solution I am going to be using permutation and combinations, the formula for

nPr is = (n!)/(n-r)!

The formula for nCr is (n!)/(r!)(n-r)!

1)

Number of upper case letters = 26

Number of lower case letters = 26

Number of number characters = 10

Plus one pound sign can be used.

a) We cannot use number characters -

Total characters we can use = 26 (Upper case) + 26 (lower case) + 1 (pound sign) = 53

So we need to place 53 characters in 8 places, where order matters -

This comes out to be 53P8 = 53*52*51*50*49*48*47*46 = 35736531667200

b)

We need to place pound sign but not at first or last position.

Total ways to place pound sign - 6

In remaining 7 positions, we need to place letters (52) + numbers (10) = 62.

Total ways to arrange in remaining positions = 62P7 = 62*61*....*56 = 2478652606080

Total ways to arrange all 8 characters = 6 * 62P7 = 14871915636480

c) Atleast one number needs to be present =

Number of ways to create using all characters - Number of ways to create using no number

= 63P8 - 35736531667200 (Answer from part a)

= 120418582515840

2)

a) To have total of 6 fruits while having atleast as many pineapples as bananas, Alison can have -

3 pineapples, 3 bananas

4 pineapples, 2 bananas

5 pineapples, 1 bananas

6 pineapples, 0 bananas

Total pineapples = 15

Total bananas = 10

Total ways to select x pineapples = 10Cx

Total ways to select x bananas = 15Cx

total ways to do this , we multiply total ways to find x bananas and y pineapples (multiplication rule)

10C3*15C3 + 10C4*15C2 + 10C5*15C1 + 10C6*15C0

= 80640 after calculating.

b)

She needs to eat 3 bananas and 3 pineapples, but it is given that she needs to eat the pineapple first-

So we need to eat 3 more bananas and 2 more pineapples after the first pineapple-

Number of ways to select the first pineapple - 15 (as there are 15 pineapples we have 15 choices)

Remaining 14 pineapples we select 2 more = 14C2 ways.

From 10 bananas we select 3 = 10C3 ways

Total ways = 15*14C2*10C3 = 163800