Pascal’s Triangle

5.1 Prove that the Pascal Triangle is symmetric with respect to the vertical line through its apex.

5.2 Prove that each row in the Pascal Triangle starts and ends with 1.

5.3 Give a proof of the formula in exercise 4.2

1 + (1 n) + (2 n) + \dots + (n - 1 n) + (n n) = 2^{n}

along the same lines. (One could expect that, similarly as for the “alternating” sum, we could get a nice formula for the sum obtained by stopping earlier, like $(0 n) + (1 n) + \dots + (k n)$ . But this is not the case: no simpler expression is known for this sum in general.)

5.4 By the Binomial Theorem, the right hand side in identity (6) is the coefficient of $x^{n} y^{n}$ in the expansion of $(x + y)^{2 n}$ . Write $(x + y)^{2 n}$ in the form $(x + y)^{n} (x + y)^{n}$ , expand both factors $(x + y)^{n}$ using the binomial theorem, and then try to figure out the coefficient of $x^{n} y^{n}$ in the product. Show that this gives another proof of identity (6).

5.5 Prove the following identity:

(0 n) (k m) + (1 n) (k - 1 m) + (2 n) (k - 2 m) + \dots + (k - 1 n) (1 m) + (k n) (0 m) = (k n + m) .

You can use a combinatorial interpretation of both sides, similarly as in the proof of (6) above, or the Binomial Theorem as in the previous exercise.

5.6 Suppose that you want to choose a $(k + 1)$ -element subset of the $(n + k + 1)$ -element set ${1, 2, \dots, n + k + 1}$ . You decide to do this by choosing first the largest element, then the rest. Show that counting the number of ways to choose the subset this way, you get a combinatorial proof of identity (7).

5.7 For which values of $n$ and $k$ is $(k + 1 n)$ twice the previous entry in the Pascal Triangle?

5.8 Instead of the ratio, look at the difference of two consecutive entries in the Pascal triangle:

(k + 1 n) - (k n) .

For which value of $k$ is this difference largest?

5.9 (a) Give a combinatorial argument to show that $2^{n} > (4 n)$ .

(b) Use this inequality to show that $2^{n} > n^{3}$ if $n$ is large enough.

5.10 Show how to use the Stirling Formula to derive (9).

5.11 Find a number $c$ such that if $t > c m$ , then

(m - t 2 m) < \frac{1}{100} (m 2 m) .

5.12 (a) Prove Lemma 5.1, by following the proof of Theorem 5.1.

(b) Show that Lemma 5.1 follows from Theorem 5.1, if one observes that as $s$ increases, the binomial coefficient $(m - t - s 2 m)$ decreases faster than the binomial coefficient $(m - s 2 m)$ .

(c) Show that by doing the calculations more carefully, the lower bound of $t^{2} / m$ in Lemma 5.1 can be improved to $1 + t (2 t + s) / m$ .

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