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?
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?