# There are 35 numbers in the Massachusetts Mass Cash game. In how many ways can a player select five of the numbers?

Apr 23, 2017

Condition where order matters$\to 38955840$
Condition where order not matter $\to 324632$

#### Explanation:

Note that the notation ! means 'Factorial'.

An example 4!->4xx3xx2xx1 where the 1 has no effect.

Case 1
If the order matters then you use the standardised form of (n!)/((n-r)!)

Case 2
If order does not matter then we use (n!)/((n-r)!r!)

The trick is to look for values you can cancel out.
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Suppose we have Case 2

Then (n!)/((n-r)!r!)" "->" "(35!)/((35-5)!5!)

Note that (35-5)! ->30!

(35xx34xx33xx32xx31xxcancel(30!))/ (cancel((30!))xx5xx4xx3xx2xx1)

$\frac{35}{5} \times \frac{34}{4} \times \frac{33}{3} \times \frac{32}{2} \times 31$

$\frac{35}{5} \times 34 \times \frac{33}{3} \times \frac{32}{4 \times 2} \times 31$

$7 \times 34 \times 11 \times 4 \times 31 = 324632$

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Suppose we have Case 1

Case 2 has part way answered it but we need to cancel out the division by 5! So we multiply by 5!

324632xx5! = 38955840