Question

Problem 6: (Permutations and Combinations). Solve the following problems in combinatorics.

a. In a card game with 52 playing cards and 4 players, you decide to split the cards into four equal parts and divide among the players. In how many different ways can you distribute the cards?

b. In a coin experiment involving tossing of 10 coins (where each 10-length sequence is equally likely), what is the probability that the outcomes 2, 3, 5, 7 are Heads?

c. You have 20 books of two subjects, with 10 books for each subject. In how many different ways can you arrange the books side-by-side on your book shelf so that no two books of the same subject are next to each other?

Answer 1

a) There are total 52 cards and we have to divide among four people.

Number of ways of dividing 52 cards equally among 4 players =

= 1287473706371731028141698560000

b) Total possible outcomes =

Total Favourable outcomes =

The probability that the outcomes 2, 3, 5, 7 are Heads = 1 / ( 2 x 2 x 2 x 2 ) = 0.0625

c)

Given there are 10 books of each subject .

Let | represent book one and * book two then we can arrange them in only two ways to satisfy the condition

|*|*|*|*|*|*|*|*|*|*

*|*|*|*|*|*|*|*|*|*|

Number of different ways to arrange the books side-by-side on the book shelf so that no two books of the same subject are next to each other = 2