John has 3 hats, 5 shirts, 3 pairs of pants and 6 pairs of socks. a) How many di

Question

John has 3 hats, 5 shirts, 3 pairs of pants and 6 pairs of socks. a) How many different ways can John dress himself (assuming nobody can see his undies and he always wears the same shoes)? b) More specifically, he has a red, blue and black hat, 2 red shirts and 3 blue shirts, 2 blue pants and one pair of black pants, 2 red socks, 2 blue socks and 2 black socks. i) How many ways can he wear all blue? All black? All red? ii) If he dresses himself at random, what is the probablilty that his hat and shirt will be different colors? A computer hacker obtains an encrypted file. She does not know the password needed to decrypt it, but through other means is able to determine that the password contains the letter o 3 times, g 3 times, m 1 time and r 1 time. a) If it takes her 3 sec to try each possible password, how long will she need to crack the file? b) How many possibilities for the password would there be if she only knows that it contains 3 of one letter, 3 of another letter, 1 of another letter and 1 of another letter (assume they are all lowercase), but she doesn't know what the letters are?

Explanation / Answer

3)

John can cjoose hat in 3 ways, a shirt in 5 ways, a pair of pant in 3 ways and a pair of socks in 6 ways so by the fundamental priciple of counting the required number of ways is

3*5*3*6 = 270 ways

b)(i)

All blue:

Number of ways of choosing blue hat is one, shirt is 3, pant is 2 and socks is 2 so number of ways of wearing all blue is 1*3*2*2 = 12.

All red:

John do not have red pant so he cannot wear all red.

All black:

John do not have black shirt so he cannot wear all black.

(ii)

That is (red shirt and blue or black hat) or (blue shirt and red or black hat).

He can choose red shirt in 2 ways and hat in 2 ways (either blue or black), pant in 3 ways and socks in 6 ways so number of ways of choosing red shirt and blue or black hat is 2*2*3*6=72

He can choose blue shirt in 3 ways and hat in 2 ways (either red or black), pant in 3 ways and socks in 6 ways so number of ways of choosing red shirt and blue or black hat is 3*2*3*6=108

So possible number of ways is 72+108 = 180

The required probability is 180 / 270 = 0.6667

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