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Chapter 5: Problem 400

Nandan gives dinner party to six guests. The number of ways in which they may be selected from ten friends if two of the friends will not attend the party together is: (a) 112 (b) 140 (c) 164 (d) 146

Short Answer

Expert verified
The short answer to the question is that there are 140 ways to invite 6 guests to Nandan's dinner party given the conditions (b) 140.

Step by step solution

01

Case 1: A attends, but B doesn't

Out of 10 friends, if A attends, then B cannot attend. So, we have 8 remaining friends to choose from (excluding A and B). Since we need 6 guests, we only need to invite 5 more from the remaining 8 friends. This can be done in \( \binom{8}{5} \) ways.
02

Case 2: B attends, but A doesn't

This case is similar to case 1. If B attends, then A cannot attend. So, we have 8 remaining friends to choose from (excluding A and B). Since we need 6 guests, we only need to invite 5 more from the remaining 8 friends. This can be done in \( \binom{8}{5} \) ways.
03

Case 3: Neither A nor B attends

If neither A nor B attend, we need to select all 6 guests from the remaining 8 friends. This can be done in \( \binom{8}{6} \) ways.
04

Total number of ways of selecting 6 guests

The total number of ways to invite 6 guests, as calculated above is the sum of the ways from each case: Total ways = (ways for case 1) + (ways for case 2) + (ways for case 3) Total ways = \( \binom{8}{5} \) + \( \binom{8}{5} \) + \( \binom{8}{6} \) Using the values of combinations, we can calculate the total ways as: Total ways = 56 + 56 + 28 = 140 Therefore, the correct answer is (b) 140.

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