Question

1. How many sets of answers are possible for a true or false quiz of 50 questions?
2.From a deck of cards, in how many ways can you pick the following cards. Ace, King, Queen and Jack.
A. If each of the pair can belong to any suit
B. If four must belong to different suits
C. If the Four must belong to the heart suit
D. If four must belong to the same suit

Answer 1

1. For each true/false question, there are two possible answers. The answer for one question has no effect on the answer for another (they are independent events).

The number of different ways that 50 questions can be answered is

$$2^{50}=1125899906842624$$

2.

- There are 52 cards in a standard deck of cards.

- There are 4 of each card (4 Aces, 4 Kings, 4 Queens, 4 Jacks, etc.)

- There are 4 suits (Clubs, Hearts, Diamonds, and Spades) and there are 13 cards in each suit

(Clubs/Spades are black, Hearts/Diamonds are red).

- Without replacement means the card IS NOT put back into the deck. With replacement means the

card IS put back into the deck. 

There are $C_{52,5}$ different possible 4-card

$$C_{52,4}=\binom{52}{4}={52! \over 4!(52-4)!}={52\cdot51\cdot50\cdot49 \over 1\cdot2\cdot3\cdot4}=270725$$

We can pick 1 Ace from 4 Aces in $C_{4,1}=4$ ways.

We can pick 1 King from 4 Kings in $C_{4,1}=4$ ways.

We can pick 1 Queen from 4 Queens in $C_{4,1}=4$ ways.

We can pick 1 Jack from 4 Jacks in $C_{4,1}=4$ ways.

A. If each of the pair can belong to any suit. 

We can pick  Ace, King, Queen and Jack in $4\cdot4\cdot4\cdot4=256$ ways.

$$p={256 \over 270725}\approx0.00094561$$

B. If four must belong to different suits. 

We can pick  Ace, King, Queen and Jack in $4!=24$ ways.

$$\begin{matrix} Clubs & Hearts & Diamonds & Spades \\ A & K & Q & J \\ A & K & J & Q \\ A & Q & K & J \\ A & Q & J & K \\ A & J & Q & K \\ A & J & K & Q \\ K & A & Q & J\\ K & A & J & Q\\ K & Q & A & J\\ K & Q & J & A \\ K & J & Q & A\\ K & J & A & Q \\ Q & A & K & J\\ Q & A & J & K\\ Q & K & A & J\\ Q & K & J & A \\ Q & J & K & A\\ Q & J & A & K \\ J & A & Q & K\\ J & A & K & Q\\ J & Q & A & K\\ J & Q & K& A \\ J & K & Q & A\\ J & K & A & Q \end{matrix}$$ $$p={24 \over 270725}\approx0.00008865$$

C. If the Four must belong to the heart suit. 

We can pick  Ace, King, Queen and Jack in $1$ way:

$$p={1 \over 270725}\approx0.00000369$$

D. If four must belong to the same suit.

We can pick  Ace, King, Queen and Jack in $4$ ways:

$$p={4 \over 270725}\approx0.00001478$$