parveen110 wrote:What's wrong with the following approach:
# of ways to select 4 ppl from:
4 single men= 4C4=1
3 single men and any one from married couple= 4C3*6C1 = 24
2 single men and any two from married couple= 4C2*6C1*4C1= 144
The portion in red is incorrect.
If your intention is to select a COMBINATION of 2 married people who are not married to each other, then the ORDER of the 2 married people doesn't matter.
Thus, we must divide 6C1*4C1 by the the number of ways the 2 married people can be arranged (2!):
4C2 * (6C1*4C1)/2! = 72.
1 single men and any three from married couple= 4C1*6C1*4C1*2C1= 192
The portion in red is incorrect.
If your intention is to select a COMBINATION of 3 married people who are not married to each other, then the ORDER of the 3 married people doesn't matter.
Thus, we must divide 6C1*4C1*2C1 by the the number of ways the 3 married people can be arranged (3!):
4C1 * (6C1*4C1*2C1)/3! = 32.
Also, accounting for at most 1 married couple= 3C2*(8C2-2) = 78
Combining:
1+24+144+192+78 which is way too much than required.
Adding together the revised totals, we get:
1+24+72+32+78 = 207.
Please help.
Thank you.
Please see my notes in red above.
The following approach is similar to yours but perhaps a bit more straightforward:
Case 1: 4 single men
Number of ways to choose 4 single men from 4 options = 4C4 = (4*3*2*1)/(4*3*2*1) = 1.
Case 2: 3 single men and 1 married person
Number of ways to choose 3 single men from 4 options = 4C3 = (4*3*2)/(3*2*1) = 4.
Number of ways to choose 1 married person from 6 options = 6C1 = 6/1 = 6.
To combine these options, we multiply:
4*6 = 24.
Case 3: 2 single men and 2 married people
Number of ways to choose 2 single men from 4 options = 4C2 = (4*3)/(2*1) = 6.
Number of ways to choose 2 married people from 6 options = 6C2 = (6*5)/(2*1) = 15.
To combine these options, we multiply:
6*15 = 90.
Case 4: 1 single men and 3 married people
Number of ways to choose 1 single man from 4 options = 4C1 = 4/1 = 4.
Number of ways to choose 3 married people from 6 options = 6C3 = (6*5*4)/(3*2*1) = 20.
To combine these options, we multiply:
4*20 = 80.
Case 5: 4 married people
Number of ways to choose 4 married people from 6 options = 6C4 = 15.
Of these 15 combinations, we must subtract those consisting of 2 married couples.
Number of ways to choose 2 married couples from 3 options = 3C2 = 3.
Subtracting the 3 disallowed combinations, we get:
15-3 = 12.
Adding together the 5 cases, we get:
1 + 24 + 90 + 80 + 12 = 207.
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