Question

In a deck of sixty special cards, 36 cards have four copies each (so there are 9 different cards of this type C1, C2, . . . , C9, and four of each in the deck, adding to 36), while the remaining 24 "L" cards are identical.
(1) How many distinguishable shuffles are there?
(2) Draw four cards. How many possible (unordered) hands of four cards are there?

Answer 1

Answer:

(1) 60

(2) 487635

Step-by-step explanation:

(1) Distinguishable shuffles means that all the cards must be different from each other. We have 9 different cards types having 4 cards each and 1 card having 24 copies of it. So, in a distinguishable shuffle, we can get 1 card each from the 9 different cards (C1 - C9) and another card from the set of 24 identical cards. So we will have a total of 10 cards.

The possible ways of getting these cards are:

9 x (⁴C₁) + ²⁴C₁

(9 x 4) + 24 = 60

There are 60 distinguishable shuffles.

(2) Here we are simply asked to calculate the number of 4-card combinations that can be drawn from the deck of 60 cards. It is not mentioned that the cards must be distinguishable so,

Possible hands of four cards = ⁶⁰C₄ = 487635