Math Problems Vault

Problem 2.1

What is the number of subsets of an $n$ -element set?

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Problem 2.2

In how many ways can you order an $n$ -element set?

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Problem 2.3

What is the number of all $k$ -element subsets of an $n$ -element set?

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Problem 2.4

What is the number of all one-to-one mappings from an $n$ -element set to an $m$ -element set?

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Problem 2.5

What is the number of all mappings from an $n$ -element set to an $m$ -element set?

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Problem 2.6

Use the above problems to justify the following particular case of Newton's binomial formula

2^{n}=\sum _{k=0}^{n}\binom {n}{k}

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Problem 2.7

Prove the Van der Monde formula

\binom {m+n}{k}=\sum _{i=0}^{k}\binom {m}{i}\binom {n}{k-i} .

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Problem 2.8

Ten people are randomly seated at a round table. What is the probability that a particular couple will sit next to each other?

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Problem 2.9

If boys and girls are born equally likely, what is the probability that in a family with three children, exactly one is a girl.

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Problem 2.10

(a)(b)(c)(d)

Two dice are thrown. What is the probability that the total number of dots is

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(a)

equal to 7,

(b)

equal to 3,

(c)

greater than 5,

(d)

an even number.

Problem 2.11

(a)(b)(c)

In a lottery, 6 numbers are drawn out of 49. Find the probability that

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(a)

$1,2,3,4,5,6$ are drawn,

(b)

$4,23,24,35,40,45$ are drawn,

(c)

44 is one of the numbers drawn.

Problem 2.12

What is the probability that among 25 people, at least 2 have their birthday on the same day of the year.

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Problem 2.13

Among $t=60$ lottery tickets, $w=20$ win prizes. We buy $b=6$ . What is the probability that $g=2$ will be winning? Generalize this to arbitrary numbers $t, w, b, g$ .

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Problem 2.14

In a series of 1000 light bulbs, $2\%$ are defective. What is the probability that among 20 bulbs bought, there are 2 faulty ones?

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Problem 2.15

From a bridge deck of 52 cards, we draw 13. What is the probability that we have 5 spades in our hand?

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Problem 2.16

A bridge deck of 52 cards is dealt among the players. Suppose that I have 4 spades and the opponents have shown (by bidding) that they have 8 hearts. What is the probability that my partner has at least 3 spades?

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Remark.

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For bridge players. This problem has practical consequences. Suppose that as South you have

\text{ * AJ84 } \bigcirc 32 \diamond 54 \text{ \& KQ952 }

and the bidding is

N	E	S	W
	$1 \mathcal{O}$	pass	$2 \mathcal{O}$
pass	pass	$?$

The risk of takeout double is $3 \diamond$ response. The best contract may be 2 中, and the estimation of the probability of a complete misfit (partner holding less than three spades) is important.

Problem 2.17

We draw 5 cards out of a deck of 24. What is the probability that we have three of one kind.

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Problem 2.18

(a)(b)

In the game of poker played with the 24-card deck you get AAAKJ.

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(a)

What is the probability of getting one Ace if you discard KJ?

(b)

What is the probability of getting a King or an Ace if you discard J?

Problem 2.19

(a)(b)

In the game of poker played with the 24-card deck you get AAKJ9.

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(a)

What is the probability of getting one Ace or two Aces if you discard KJ9?

(b)

What is the probability of getting a King or an Ace if you discard J?

Problem 2.20

A monkey hits a computer keyboard three times at random. What is the chance of getting a three-letter word with a consonant followed by two vowels? The word does not have to make sense. For simplicity, assume that there are 100 keys.

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Problem 2.21

From a pack of 52 cards, we draw one-by-one. What is the probability that an Ace will appear for the first time at the fifth turn?

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Problem 2.22

How likely is it that the word ABRACADABRA will show if the letters $\mathrm{A}, \mathrm{A}, \mathrm{A}, \mathrm{A}, \mathrm{A}, \mathrm{B}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{R}, \mathrm{R}$ are shuffled randomly?

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