Math CentralQuandaries & Queries


Question from Ethel, a student:

This is a question from the PTSP Success Test. There are 4 horses at a stable. And how many different orders can they be ridden by their owners? The answer choices are A. 4 B. 10 C. 16 D. 24 and E. 36

Hi Ethel,

Suppose the horses are Bob, Carole, Ted and Alice. Think about making a list of all possible orders. There are four choices for the first horse ridden.


Now, whichever horse you chose first you have three choices for which horse to ride next

Bob, Carole
Bob, Ted
Bob, Alice
Carole, Bob
Carole, Ted
Carole, Alice
Ted, Bob
Ted, Carole
Ted, Alice
Alice, Bob
Alice, Carole
Alice, Ted

Hence there are 4 × 3 = 12 ways you can choose the first two horses.

Again, whichever horses you have chosen so far you have 2 choices for the third horse to ride. Thus each of the 12 orders in the list above expands to 2 orders with three horses. For example Bob, Carole becomes Bob, Carole, Ted and Bob, Carole, Alice. Hence there are 4 × 3 × 2 = 24 possible orders you can choose to ride 3 of the horses. Finally if you have chosen the order for the first three there is only one horse to choose as the last horse to ride. Thus in total there are 4 × 3 × 2 × 1 = 24 possible orders in which to choose the horses.


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