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Number of ways of forming 10 student committee from 5 classes of 30 students each

There are 5 classes with 30 students each. How many ways can a committee of 10 students be formed if each class has to have at least one student on the committee?

Let $$X, Y, Z, U, V$$ be five sorts of students

Then $$ Reps(X) = X \cdot \mathit{29}(X) + \mathit2(X) \cdot \mathit{28}(X) + \mathit3(X) \cdot \mathit{27}(X) + \mathit4(X) \cdot \mathit{26}(X) + \mathit 5(X) \cdot \mathit{25}(X) + \mathit6(X) \cdot \mathit{26}(X) $$

reprezents possible choices for the representants of the first class.

Passing to e.g.f. and adding a counter $$t$$ for the number of cosen students from one class, one gets:

$$ a = t { x \over 1!} {x^{29} \over 29!} + t^2 { x^2 \over 2!} {x^{28} \over 28!} + \dots + t^6 { x^6 \over 6!} {x^{24} \over 24!}  $$

$$ = ( { 30!  \over 1! 29!}t           + { 30!  \over 2! 28!}t^2           + \dots +  { 30!\over 6! 24!})t^6 \frac {x^{30}} {30!} $$

Similarly we get

$$ b = ( { 30!  \over 1! 29!}t           + { 30!  \over 2! 28!}t^2           + \dots +  { 30!\over 6! 24!})t^6 \frac {y^{30}} {30!} $$

and so on.

By multiplying the obtained $$a(t,x)b(t,y) \cdots e(t,v) $$ and then taking the coefficient of

$$t^{10} \frac {x^{30}} {30!} \cdots \frac {v^{30}} {30!} $$

one gets $$645666069796875$$.