Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 2 / a

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- Solution a

Solution

Count pairs

(M, K)

of subsets of an

n

-element candidate set with

K \subseteq M

,

∣ M ∣ = m

, and

∣ K ∣ = k

. Choosing the whole team

M

first and then the special-role players

K

gives

(m n) (k m)

. Alternatively choose

K

first, then choose the other

m - k

players from the

n - k

candidates outside

K

(k n) (m - k n - k)

. Each construction specifies the same pair uniquely. Thus double counting proves

(m n) (k m) = (k n) (m - k n - k) .

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