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John has 7 relatives, 3 men and 4 women. His wife Karen also has 7 relatives 3 women and 4 men. In how many ways can they invite 3 men and 3 women so that 3 of them are John's relatives and 3 his wife Karen's?
A. 485
B. 365
C. 458
D. 356
E. 285
A. 485
B. 365
C. 458
D. 356
E. 285
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12 Mar 2017, 21:38
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There are 4 cases:
John | Karen
relatives | relatives
Case invited | invited
----------------------------------
1. 3M, 0W | 0M, 3W
2. 2M, 1W | 1M, 2W
3. 1M, 2W | 2M, 1W
4. 0M, 3W | 3M, 0W
Case 1: Choose the John 3 men relatives 3C3 = 1 way.
Choose the Karen 3 women relatives 3C3=1 ways.
That's 1*1 = 1 way for Case 1.
Case 2: Choose the John 2 men relatives 3C2 = 3 ways.
Choose the Karen 1 woman relative 4C1 = 4 ways.
Choose the Karen 1 man relative 4C1=4 ways.
Choose the Karen 2 women relatives 3C2 = 3 ways.
That's 3*4*4*3 = 144 ways for Case 2.
Case 3: Choose the John 1 man relative 3C1 = 3 ways.
Choose the John 2 women relatives 4C2 = 6 ways.
Choose the Karen 2 men relative 4C2 = 6 ways.
Choose the Karen 1 woman relative 3C1 = 3 ways.
That's 3*6*6*3 = 324 ways for Case 3.
Case 4: Choose the John 3 women relatives 4C3 = 4 ways.
Choose the Karen 3 men relatives 4C3=4 way.
That's 4*4 = 16 ways for Case 4.
Grand total from all cases: 1+144+324+16 = 485 ways.
John | Karen
relatives | relatives
Case invited | invited
----------------------------------
1. 3M, 0W | 0M, 3W
2. 2M, 1W | 1M, 2W
3. 1M, 2W | 2M, 1W
4. 0M, 3W | 3M, 0W
Case 1: Choose the John 3 men relatives 3C3 = 1 way.
Choose the Karen 3 women relatives 3C3=1 ways.
That's 1*1 = 1 way for Case 1.
Case 2: Choose the John 2 men relatives 3C2 = 3 ways.
Choose the Karen 1 woman relative 4C1 = 4 ways.
Choose the Karen 1 man relative 4C1=4 ways.
Choose the Karen 2 women relatives 3C2 = 3 ways.
That's 3*4*4*3 = 144 ways for Case 2.
Case 3: Choose the John 1 man relative 3C1 = 3 ways.
Choose the John 2 women relatives 4C2 = 6 ways.
Choose the Karen 2 men relative 4C2 = 6 ways.
Choose the Karen 1 woman relative 3C1 = 3 ways.
That's 3*6*6*3 = 324 ways for Case 3.
Case 4: Choose the John 3 women relatives 4C3 = 4 ways.
Choose the Karen 3 men relatives 4C3=4 way.
That's 4*4 = 16 ways for Case 4.
Grand total from all cases: 1+144+324+16 = 485 ways.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
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