Question

If a coin is tossed 55 times, and then a standard six-sided die is rolled 22 times, and finally a group of five cards are drawn from a standard deck of 5252 cards without replacement, how many different outcomes are possible

Answer 1

2,994,001,920 different outcomes are possible.

Explanation:

The coin, the die and the cards are independent events, so the fundamental counting principle is used.

Also, the order in which the cards are chosen is not important, which means that the combinations formula is used to solve this question.

Fundamental counting principle:

States that if there are p ways to do a thing, and q ways to do another thing, and these two things are independent, there are p*q ways to do both things.

Combinations formula:

`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

Coin:

Tossed 5 times, each toss 2 possible outcomes, so:

`2^5 = 32`

32 possible outcomes.

Die:

Tossed 2 times, each toss with 6 possible outcomes, so:

`6^2 = 36`

36 possible outcomes.

Cards:

Chosen without replacement, so the combinations formula is used, 5 from a set of 52. So

`C_(52,5) = (52!)/(5!47!) = 2598960`

How many different outcomes are possible?

Using the fundamental counting principle:

`T = 32*36*2598960  = 2994001920`

2,994,001,920 different outcomes are possible.