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

1 Answer
Apr 23, 2017

Answer:

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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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#