The Super Lotto Plus game consists of a player selecting fiv

The Super Lotto Plus game consists of a player selecting five different numbers from 1 to 47 and a mega number

from 1 to 27 (the mega number need not be different from the other numbers). (a) How many different selections

are possible? (b) How many ways can a player select numbers such that three of the numbers match and the mega

number doesn

Solution

Part (a)

Step 1

As stated the first five numbers can be from 1 to 47

This selection can be made in ⁴⁷C_⁵ ways or 1,5333,939

Step 2

The mega number is selected from numbers 1 to 27

This selection can be made in ²⁷C₁ ways or 27

So the two selections together can be made in ⁴⁷C₅ X ²⁷C₁ ways.

Part (b)

Step 1

Three of the first five selections made are such that three of these numbers match (between (1 to 27) while the other two do not match (between 28 to 47)

These selections can be made in ²⁷C _³X 20C2 ways

Step 2

The mega number selected is such that it doesnt match any of the numbers selected means that out of 27 since three have already been selected, the mega number would be selected form the remaining 24 numbers

This can be done in ²⁴C₁ ways

The two selections together can be made in ²⁷C ₃ X ²⁰C₂ X ²⁴C₁ ways

Part (c)

Step 1

The five selections are made in such a way that none of the five numbers match meaning that they would be numbers from 28 to 47 ( as 1 -27 match for the two draws)

This can be done in ²⁰C₅ ways

Step 2

The mega number would match the numbers drawn if it is selected such that it is not between 1 to 27

This can be doen in ²⁰C₁ ways

The two selections together would be done in ²⁰C₅ ways X _²⁰C1 ways

Note: All calculations would be done using nCr = n!/r!(n-r)!