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1234Combinations

Solutions to the exercises

Exercise 1

$(\begin{matrix} 10 \\ 3 \end{matrix}) = 120$

$(\begin{matrix} 10 \\ 9 \end{matrix}) = 10$

$2^{10} = 1024$

Exercise 2

The outcomes can be 0, 1, 2, 3, $4$ or $5$ times "head". Therefore there are $6$ possibilities.

$(\begin{matrix} 5 \\ 2 \end{matrix}) = 10$

$(\begin{matrix} 50 \\ 20 \end{matrix}) \approx 4,71 \cdot 10^{13}$

Exercise 3

For each game two participants are selected from $24$ and the order is not relevant. So there are $(\begin{matrix} 24 \\ 2 \end{matrix}) = 276$ games to be played.

Exercise 4

$(\begin{matrix} 14 \\ 4 \end{matrix}) = 1001$

$(\begin{matrix} 14 \\ 2 \end{matrix}) \cdot (\begin{matrix} 12 \\ 2 \end{matrix}) = 6006$

Exercise 5

$8! = 40320$

$5! \cdot 3! = 720$

$6! \cdot 2 = 1440$

$(\begin{matrix} 8 \\ 3 \end{matrix}) \cdot 5! = 6720$

Exercise 6

$6 \cdot 6 \cdot 6 = 216$

$19$

Exercise 7

$(\begin{matrix} 18 \\ 4 \end{matrix}) = 3060$

$(\begin{matrix} 9 \\ 2 \end{matrix}) \cdot (\begin{matrix} 9 \\ 2 \end{matrix}) = 1296$

$18 \cdot 17 \cdot 16 \cdot 15 = 73440$

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