there are 100 students in a school. it is found out that each student should select at least 4 courses, so that no two students have the same selection. how many different courses does the school offer?


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Question Number 218445 by mr W last updated on 10/Apr/25

$t h e r e a r e 100 s t u d e n t s i n a s c h o o l .$ $i t i s f o u n d o u t t h a t e a c h s t u d e n t$ $s h o u l d s e l e c t a t l e a s t 4 c o u r s e s, s o$ $t h a t n o t w o s t u d e n t s h a v e t h e s a m e$ $s e l e c t i o n .$ $h o w m a n y d i f f e r e n t c o u r s e s d o e s$ $t h e s c h o o l o f f e r ?$
	Answered by MrGaster last updated on 10/Apr/25
	$C:∪_(k=4) ^v ((([v])),(k) )⊇{S_i }_(i=1) ^(100) ,S_i ⊆[v],∣S_i ∣≥4,S_i ≠S_j ∀i≠j min v s.t∣∪_(k=4) ^v ((([v])),(k) )∣≥100 ⇒Σ_(k=4) ^v ((v),(k) )≥100 Lower bound via ((v),(4) )≤Σ_(k=4) ^v ((v),(4) )<2^v ((v),(4) )=((v(v−1)(v−2)(v−3))/(24))≥100 ⇒v(v−1)(v−2)(v−3)≥2400 Test v=9: 9×8×7×6=3024≥2400 Test v=8: 8×7×6×5=1680<2400 ⇒ determinant ((9))$
	$C : \underset{k = 4}{\overset{v}{\cup}} (\begin{matrix} [v] \\ k \end{matrix}) \supseteq {S_{i}}_{i = 1}^{100}, S_{i} \subseteq [v], ∣ S_{i} ∣\geq 4, S_{i} \neq S_{j} \forall i \neq j$ $\min v s . t ∣ \underset{k = 4}{\overset{v}{\cup}} (\begin{matrix} [v] \\ k \end{matrix}) ∣\geq 100$ $\Rightarrow \underset{k = 4}{\sum^{v}} (\begin{matrix} v \\ k \end{matrix}) \geq 100$ $Lower bound via (\begin{matrix} v \\ 4 \end{matrix}) \leq \underset{k = 4}{\sum^{v}} (\begin{matrix} v \\ 4 \end{matrix}) < 2^{v}$ $(\begin{matrix} v \\ 4 \end{matrix}) = \frac{v (v - 1) (v - 2) (v - 3)}{24} \geq 100$ $\Rightarrow v (v - 1) (v - 2) (v - 3) \geq 2400$ $Test v = 9 : 9 \times 8 \times 7 \times 6 = 3024 \geq 2400$ $Test v = 8 : 8 \times 7 \times 6 \times 5 = 1680 < 2400$ $\Rightarrow \begin{matrix} 9 \end{matrix}$
	Commented by mr W last updated on 10/Apr/25
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	Answered by mr W last updated on 10/Apr/25

	$s a y t h e s c h o o l o f f e r s n c o u r s e s .$ $C_{n}^{4} ⩾ 100 \land C_{n}^{3} < 100$ $\Rightarrow n = 9$
	Answered by Spillover last updated on 12/Apr/25

	n!/4!/(n-4)!=100 n!/(n-4)!=2400 n(n-1)(n-2)(n-3)=2400 8<n<9 N=9
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