<p>Nine individuals - Z, Y, X, W, V, U, T, S and R - are the only candidates, who can serve on three committees-K1, K2 and K3, and each candidate should serve on exactly one of the committees. Committee K1 should consist of exactly one member more than committee K2. It is possible that there are no members in committee K3. Among Z, Y and X none can serve on committee 10. Among W, V and U none can serve on committee K2. Among T, S and R none can serve on committee K3. In case committee K2 is served by T and Z only, how many of the nine individuals should serve on committee K3? </p> <p>Of the nine individuals, the largest number that can serve together on committee K3 is:</p>
Step-by-step Solution:
To find the largest possible number of members for Committee K3, we must first find all mathematically possible sizes based on the size rules, and then check which of those are actually possible given the rules about who can serve on which committee.
We are given two rules regarding the number of members (size) in each committee:
By substituting the first rule into the second, we can create a single equation relating the sizes of K2 and K3:
(Size(K2) + 1) + Size(K2) + Size(K3) = 9
2 × Size(K2) + Size(K3) = 8
From this equation, to maximize Size(K3), we must minimize Size(K2). The smallest possible size for K2 is 0. If Size(K2) = 0, then Size(K3) = 8. Other possible integer values for Size(K3) are 6, 4, 2, and 0.
Now we must consider which individuals are allowed to serve on Committee K3.
This creates a new constraint: Size(K3) ≤ 6.
The mathematical possibilities for the size of K3 were {0, 2, 4, 6, 8}. However, the membership rule limits the size to a maximum of 6. This eliminates the possibility of K3 having 8 members.
The largest possible size for K3 that satisfies both the size equation and the membership rule is therefore 6.
We can verify that a valid assignment exists for Size(K3)=6:
The largest number of individuals that can serve together on committee K3 is 6.