Counting subsets

4.1 Illustrate this argument by a tree.

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4.2 Suppose that we record the order of all 100 athletes.

(a) How many different outcomes can we have then?

(b) How many of these give the same for the first 10 places?

(c) Show that the result above for the number of possible outcomes for the first 10 places can be also obtained using (a) and (b).

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4.3 If you generalize the solution of exercise 4.1, you get the answer in the form

$$\frac{n!}{(n-k)!}$$

Check that this is the same number as given in theorem 4.1.

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4.4 Explain the similarity and the difference between the counting questions answered by theorem 4.1 and theorem 2.2.

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4.5 Which problems discussed during the party were special cases of theorem 4.2?

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4.6 Tabulate the values of $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ for $n,k\le 5$.

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4.7 Prove that

$$\left(\genfrac{}{}{0ex}{}{n}{2}\right)+\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)=n^2.$$

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4.8 (a) Prove that

$$\left(\genfrac{}{}{0ex}{}{90}{5}\right)=\left(\genfrac{}{}{0ex}{}{89}{5}\right)+\left(\genfrac{}{}{0ex}{}{89}{4}\right).$$

(b) Formulate and prove a general identity based on this.

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4.9 Prove that

$$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-k}\right).$$

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4.10 Prove that

$$1+\left(\genfrac{}{}{0ex}{}{n}{1}\right)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{n}{n-1}\right)+\left(\genfrac{}{}{0ex}{}{n}{n}\right)=2^n.$$

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4.11 Prove that for $0<c\le b\le a$,

$$\left(\genfrac{}{}{0ex}{}{a}{b}\right)\left(\genfrac{}{}{0ex}{}{b}{c}\right)=\left(\genfrac{}{}{0ex}{}{a}{a-c}\right)\left(\genfrac{}{}{0ex}{}{a-c}{b-c}\right).$$

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4.12 Give a proof of the Binomial Theorem by induction, based on exercise 4.2.

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4.13 (a) Prove the identity

$$\left(\genfrac{}{}{0ex}{}{n}{0}\right)-\left(\genfrac{}{}{0ex}{}{n}{1}\right)+\left(\genfrac{}{}{0ex}{}{n}{2}\right)-\left(\genfrac{}{}{0ex}{}{n}{3}\right)+\cdots =0$$

(The sum ends with $\left(\genfrac{}{}{0ex}{}{n}{n}\right)=1$, with the last depending on the parity of $n$.)

(b) This identity is obvious if $n$ is odd. Why?

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4.14 Prove identity 4, using a combinatorial interpretation of the two sides (recall exercise 4.2).

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4.15 We can describe the procedure of distributing the presents as follows. First, we select $n_1$ presents and give them to the first child. This can be done in $\left(\genfrac{}{}{0ex}{}{n}{n_1}\right)$ ways. Then we select $n_2$ presents from the remaining $n-n_1$ and give them to the second child, etc.

Complete this argument and show that it leads to the same result as the previous one.

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4.16 The following special cases should be familiar from previous problems and theorems. Explain why.

(a) $n=k$, $n_1=n_2=\cdots =n_k$;

(b) $n_1=n_2=\cdots =n_{k-1}=1$, $n_k=n-k+1$;

(c) $k=2$;

(d) $k=3$, $n=6$, $n_1=n_2=n_3=2$.

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4.17 (a) How many ways can you place $n$ rooks on a chessboard so that no two attack each other (Figure 7)? We assume that the rooks are identical, so e.g. interchanging two rooks does not count as a separate placement.

(b) How many ways can you do this if you have 4 black and 4 white rooks?

(c) How many ways can you do this if all the 8 rooks are different?

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4.18 How many anagrams can you make from the word COMBINATORICS?

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4.19 Which word gives rise to more anagrams: COMBINATORICS or COMBINATORICA? (The latter is the Latin name of the subject.)

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4.20 Which word with 13 letters gives rise to the most anagrams? Which word gives rise to the least?

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4.21 It is clear that STATUS and LETTER have the same number of anagrams (in fact, $6!/(2!2!)=180$). We say that these words are “essentially the same” (at least as far as counting anagrams goes): they have two letters repeated twice and two letters occurring only once.

(a) How many 6-letter words are there? (As before, the words don’t have to be meaningful. The alphabet has 26 letters.)

(b) How many words with 6 letters are “essentially the same” as the word LETTER?

(c) How many “essentially different” 6-letter words are there?

(d) Try to find a general answer to question (c) (that is, how many “essentially different” words are there on $n$ letters?). If you can’t find it, read the following section and return to this exercise after it.

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4.22 In how many ways can you distribute $n$ pennies to $k$ children, if each child is supposed to get at least 2?

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4.23 We distribute $n$ pennies to $k$ boys and $\ell$ girls, so that (to be really unfair) we require that each of the girls gets at least one penny. In how many ways can we do this?

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4.24 $k$ earls play cards. Originally, they all have $p$ pennies. At the end of the game, they count how much money they have. They do not borrow from each other, so that they cannot loose more than their $p$ pennies. How many possible results are there?

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