Amit has five friends: 3 girls and 2 boys. Amits wife also has 5 friends: 3 boys and 2 girls. In how many maximum number of different ways can they invite 2 boys and 2 girls such that two of them are Amits friends and two are his wifes?

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Q: 17 (IAS/2007)
Amit has five friends: 3 girls and 2 boys. Amit’s wife also has 5 friends: 3 boys and 2 girls. In how many maximum number of different ways can they invite 2 boys and 2 girls such that two of them are Amit’s friends and two are his wife’s?

question_subject: 

Maths

question_exam: 

IAS

stats: 

0,5,6,4,2,5,0

keywords: 

{'many maximum number': [0, 0, 1, 0], 'friends': [0, 0, 2, 0], 'amit': [0, 0, 1, 0], 'girls': [0, 2, 3, 10], 'boys': [0, 1, 5, 11], 'wife': [2, 3, 2, 2], 'different ways': [0, 0, 5, 0]}

The question is about the number of ways the invitations can be distributed among boys and girls in different ways. Amit and his wife have 2 possible patterns of inviting friends: Amit can invite 2 boys and the wife invites 2 girls, or Amit can invite 2 girls and his wife invites 2 boys.

In the first case, Amit can invite 2 boys from 2 in C(2,2) = 1 way and his wife can invite 2 girls from 2 in C(2,2) = 1 way. And in the second case, Amit can invite 2 ladies from 3 in C(3,2) = 3 ways, and his wife can invite 2 boys from 3 in "3 choose 2" ways i.e., C(3,2) = 3 ways.

Therefore total number of ways = 2* (1*1 + 3*3 ) = 2*(1+9) = 2*10 = 20.

Option 3 (46) is the provided answer but it doesn`t align with the explanation. Therefore, alert - correct answer should be none of the given options.

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