Have 5 friends in how many ways can I invite them for dinner?

Algebra -> Permutations -> SOLUTION: Help! A woman has 11 close friends. In how many ways can she invite 5 of them for dinner, if there are 2 who are not in good talking terms and so both can not be at the dinner, b Log On

Question 1146066: Help!
A woman has 11 close friends. In how many ways can she invite 5 of them for dinner, if there are 2 who are not in good talking terms and so both can not be at the dinner, but at most one of them can be invited?
Answer by VFBundy(438)

(Show Source):

You can put this solution on YOUR website!

There are 9 "regular" friends and 2 "other" friends. ("Other" friends are friends that cannot both be invited at the same time.) The woman can invite the following combinations:
5 regular friends and 0 other friends 4 regular friends and 1 other friend

Ways to invite 5 regular friends and 0 other friends:

9C5 =

= 126
Ways to invite 4 regular friends and 1 other friend:

9C4 * 2C1 =

= 126 * 2 = 252

So...there are a total of 126 + 252...or 378 ways she can do the invites.

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Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

12 Mar 2021, 02:15

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Difficulty:

25% (medium)

Question Stats:

79% (01:28) correct

21% (01:31) wrong

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Lisa has 5 friends in how many ways can she invite one or more of them at a dinner party?(A) 63(B) 32(C) 31(D) 25

(E) 24

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

24 Mar 2021, 04:57

Bunuel wrote:

Lisa has 5 friends in how many ways can she invite one or more of them at a dinner party?(A) 63(B) 32(C) 31(D) 25

(E) 24

Lisa can invite either \(1\) or \(2\) or \(3\) or \(4\) or all \(5 \) of them to the party:Hence \(5c1+ 5c2+5c3+5c4+5c5 = 5+10+10+5+1= 31\)Ans-CHope it's clear. _________________

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

29 Mar 2021, 10:09

Bunuel wrote:

Lisa has 5 friends in how many ways can she invite one or more of them at a dinner party?(A) 63(B) 32(C) 31(D) 25

(E) 24

Solution:Lisa can invite exactly r (where r = 1, 2, 3, 4, or 5) friends in 5Cr ways. Therefore, Lisa can invite one or more of her 5 friends in a total of 5C1 + 5C2 + 5C3 + 5C4 + 5C5 = 2^5 - 5C0 = 32 - 1 = 31 ways.

Alternate Solution:

If we think of Lisa’s five friends as a set, then the number of ways she can invite her friends to the dinner party is in one to one correspondence with the subsets of the set of friends. Recall that a set with five elements has 2^5 = 32 subsets. One of these subsets is the empty set, which corresponds to inviting zero friends. Taking away the empty set, we see that there are 32 - 1 = 31 ways Lisa can invite one or more of her friends.

Answer: C

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

30 Mar 2021, 06:04

Given• Lisa has 5 friends.

To find

• The number of ways Lisa can invite one or more of her friends at a dinner party.

Approach and Working out:

Method 1

Lisa can invite either 1 friend or 2 or 3 or 4 or all of them together.

Method 2Total ways of inviting at least 1 friend at the party = Total ways of inviting some friends (0, 1, 2, 3, 4, or 5) out of five friends – #ways when she does not invite any friendHence, option C is the correct answer.

Correct Answer: Option C

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

27 Apr 2022, 00:38

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Re: Lisa has 5 friends in how many ways can she invite one or more of them [#permalink]

27 Apr 2022, 00:38