Question 1161906
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In your posts, you keep showing the formulas for nPr and nCr being the same; they are not.<br>
nPr is the PERMUTATION of n things r at a time; in a permutation, order matters.  The formula for the number of permutations is<br>
nPr = n!/(n-r)!<br>
In a final race with 8 competitors, the number of ways to award first and second places (order matters) is 8P2 = 8!/6! = 8*7 = 56.<br>
nCr is the COMBINATION of n things r at a time; in a combination, order does not matter.  The formula for the number of combinations is<br>
nCr = n!/(r!(n-r)!)<br>
In a preliminary race with 8 competitors, the number of ways to select 2 of them to move to the next round (order does not matter) is 8C2 = 8!/(2!(6!)) = (8*7)/(2*1) = 28.<br>
In this problem, there is nothing that involves permutations; you are selecting a committee, so order does not matter.  So all the calculations involve nCr.<br>
As for your wrong answer, most if not all of the reason is that there are only 15 people total, not 16.  So the denominator of your probability fractions is 15C3, not 16C3.<br>
(a) 5C3/15C3  [all 3 are teachers]
(b) 4C3/15C3  [all 3 are parents]
(c) 1-10C3/15C3  [not 0 are teachers]<br>
Do the calculations again and see if you get the right answers.<br>
Re-post if you still have questions.<br>
By the way... thanks for showing the work you did.  VERY few users of this forum do that.<br>