1. Consider the sequence from the third column of Pascal's triangle. Starting with , the term of the sequence is . Prove that for all , (i) and (ii) .

(i) So . And . Thus .

(ii) Similarly, .

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 cards?

Its just ?

3. Let be a positive integer, . If , prove that .

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

For (3) would you just use the definition of multinomial coefficient: for an algebraic argument?

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

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