Pascal

1. Consider the sequence $0,0,1,3,6,10,15, \ldots$ from the third column of Pascal's triangle. Starting with $n=1$ , the $n \text{th}$ term of the sequence is $a_n = C(n,2)$ . Prove that for all $n \geq 0$ , (i) $a_{n+1}-a_n = n$ and (ii) $a_{n+1}+a_n = n^2$ .

(i) So $a_{n+1}-a_n = C(n+1,2) - C(n,2)$ . And $C(n+1,2) = C(n,1) + C(n,2)$ . Thus $a_{n+1}-a_n= C(n,1) = n$ .

(ii) Similarly, $a_{n+1}+a_n = C(n,1)+2C(n,2) = n^2$ .

Are these correct?

2. Poker is sometimes played with a joker. How many different five-card poker hands can be "chosen" form a deck of $53$ cards?

Its just $\binom{53}{5}$ ?

3. Let $r_i$ be a positive integer, $1 \leq i \leq k$ . If $n = r_1+r_2+ \cdots + r_k$ , prove that $\binom{n}{r_1,r_2, \ldots, r_k} = \binom{n-1}{r_1-1,r_2, \ldots, r_k} + \binom{n-1}{r_1,r_2-1, \ldots, r_k} + \cdots + \binom{n-1}{r_1,r_2, \ldots, r_k-1}$ .

So we can use both algebraic and combinatorial arguments for this?

Quote:

Originally Posted by Sampras

1. Consider the sequence $0,0,1,3,6,10,15, \ldots$ from the third column of Pascal's triangle. Starting with $n=1$ , the $n \text{th}$ term of the sequence is $a_n = C(n,2)$ . Prove that for all $n \geq 0$ , (i) $a_{n+1}-a_n = n$ and (ii) $a_{n+1}+a_n = n^2$ .

(i) So $a_{n+1}-a_n = C(n+1,2) - C(n,2)$ . And $C(n+1,2) = C(n,1) + C(n,2)$ . Thus $a_{n+1}-a_n= C(n,1) = n$ .

(ii) Similarly, $a_{n+1}+a_n = C(n,1)+2C(n,2) = n^2$ .

Are these correct?

2. Poker is sometimes played with a joker. How many different five-card poker hands can be "chosen" form a deck of $53$ cards?

Its just $\binom{53}{5}$ ?

3. Let $r_i$ be a positive integer, $1 \leq i \leq k$ . If $n = r_1+r_2+ \cdots + r_k$ , prove that $\binom{n}{r_1,r_2, \ldots, r_k} = \binom{n-1}{r_1-1,r_2, \ldots, r_k} + \binom{n-1}{r_1,r_2-1, \ldots, r_k} + \cdots + \binom{n-1}{r_1,r_2, \ldots, r_k-1}$ .

So we can use both algebraic and combinatorial arguments for this?

For (3) would you just use the definition of multinomial coefficient: $\binom{n}{k_{1}, k_{2}, \ldots, k_m} = \frac{n!}{k_{1}!k_{2}! \cdots k_{m}!}$ for an algebraic argument?

And for a combinatorial argument, you would consider the number of $r_1$ subsets, the number of $r_2$ subsets, etc....?

Because $C(n,r)$ is interpreted as how many ways we can choose an $r$ -subset from $n$ elements. In the same way, can we use this interpretation for the multinomial coefficient?