Another type of listing involves sequential combinations; often known as the menu problem. This is a combination of possible items, followed by a further combination of possible items.

If a burger bar was to offer a meal deal, it might offer:

 Burger Fries Cold Drink or AND or AND or Salad Rings Hot Drink

This is tackled by working down each column in turn. Use just the first letter of the menu item - making sure that there are no duplicates. Take the top entry of the first two columns (Burgers and Fries), and work down the third columns to pick up Cold and Hot drinks)

 B F C B F H

When that is done, keep the first column the same (Burgers) but move one item down on the second column (Rings). Work down the third column again.

 B F C B F H B R C B R H

Now move one item down on the first column (Salad), and repeat the whole process.

 B F C S F C B F H S F H B R C S R C B R H S R H

To obtain the total number of permutations, work out the number of items in each list, and multiply the totals for each list together. In the above example, 2 x 2 x 2 = 8.

## Example 1

Alice is going to see her friend. She will either cycle or walk to the station. From the station she will take either a railway train or a bus.

At the other end of this journey, she will either get a lift or take a taxi.

How many combinations of transport could Alice take in going to see her friend?

The total number of ways is the number of options in the first list time the total in the second times the total in the third: 2 x 2 x 2 = 8.

C-Cycle W-Walk; B-Bus; R-Railway train, L-Lift, T-Taxi

 First Stage Second Stage Third Stage Cycle Railway Lift or or or Walk Bus Taxi

And obtain these combinations:

 C R L W R L C R T W R T C B L W B L C B T W B T

## Example 2

Simon is writing a song. It will feature either a Drummer or a Bass player, but not both. It will also have either a Keyboards player, a Guitarist or a Singer.

How many different combinations of instruments/singers could Simon have in his song?

2 in the first list x 3 in the second list = 6 combinations

 Drummer Keyboards or or Bass Guitarist or Singer

And obtain these combinations:

 D K D G D S B K B G B S