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Chapter 34: Problem 1092

A man has 6 friends; in how many ways may he invite one or more of them to dinner?

Short Answer

Expert verified
The man can invite one or more of his 6 friends to dinner in 63 different ways.

Step by step solution

01

Identify the problem type

The problem is asking us to find the number of ways in which the man can invite one or more of his 6 friends to dinner, which means we are dealing with a problem of combinations.
02

Use the formula

Since the man has 6 friends, he has a possibility of inviting each friend or not inviting them. The formula for combinations in this case is: \(Number\: of\: ways = 2^n - 1 \) where n is the number of friends. Here, n = 6 and the -1 comes from the fact that he must invite at least one friend, so we need to subtract the possibility of inviting no one.
03

Calculate the number of ways

Now, we will use the formula to find the number of ways to invite one or more friends: \(Number\: of\: ways = 2^6 - 1 \)
04

Simplify the expression

\(Number\: of\: ways = 64 - 1 \newline Number\: of\: ways = 63 \)
05

Interpret the result

The man can invite his friends to dinner in 63 different ways.

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