A certain law firm consists of 4 senior partners and 6 junior partners. How many different groups of 3 partners can be formed in which at least one member of the group is a senior partner. (Two groups are considered different if at least one group member is different)
We have 4 seniors and 6 juniors.
We are asked for the nomber of groups of 3 in which at least 1 is a senior (1, 2 or 3 seniors in each group).
Considering that we have 3 types of groups:
Groups with 1 senior:
We take 1 senior out of 4 (\(C^4_1\)) and combine them with 2 juniors out of 6 (\(C^6_2\)):
\(C^4_1*C^6_2 = 4*15 = 60\)
Groups with 2 seniors:
We take 2 seniors out of4 (\(C^4_2\)) and combine them with 1 juniors out of 6 (\(C^6_1\)):
\(C^4_2*C^6_1 = 6*6 = 36\)
Groups with 3 seniors (and zero juniors):
We take 3 seniors out of 4 (\(C^4_3\)):
\(C^4_3\) = 460 + 36 + 4 = 100
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