Question

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

Answer 1

Condition where order matters`->38955840`
Condition where order not matter `->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)`

`35/5xx34/4xx33/3xx32/2xx31`

`35/5xx34xx33/3xx32/(4xx2)xx31`

`7xx34xx11xx4xx31 = 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`