Counting with Restrictions

Counting with restrictions involves counting the number of ways to arrange or select items while considering certain limitations or conditions.

Permutations with Restrictions

When arranging items with restrictions, use the following formula for permutations:

nPr = n! / (n-r)!

Where n is the total number of items and r is the number of items to arrange. This formula accounts for restrictions on the arrangement of items.

Combinations with Restrictions

When selecting items with restrictions, use the following formula for combinations:

nCr = n! / (r!(n-r)!)

Where n is the total number of items, and r is the number of items to select. This formula considers restrictions on the selection of items.

Example Problems:

1. How many ways can 5 people be arranged in a line if 2 particular people must be next to each other?

Solution:

n = 5 (total people), r = 4 (arrangement excluding the 2 particular people)

nPr = 5! / (5-4)! = 5! / 1! = 5*4*3*2*1 = 120

So, there are 120 ways to arrange the 5 people with the given restriction.

2. A committee of 3 people is to be chosen from 5 men and 4 women. How many different committees can be formed if at least 1 man must be included?

Solution:

Number of committees with at least 1 man = Total committees - Committees with no men

nCr = 9C3 - 4C3 = (9! / (3!(9-3)!)) - (4! / (3!(4-3)!)) = 84 - 4 = 80

There are 80 different committees that can be formed with the given restriction.