Medal ceremony
The teacher has forgotten which pupil won which medal. In how many different ways could he give the medals out to the pupils?
Age
14 to 16
Challenge level
Problem
6 pupils have, between them, won three gold medals, two silver medals and a bronze medal in a painting competition. Unfortunately, their teacher has lost all record of which medals should go to which pupils, so he allocates them by drawing names out of a hat. The first 3 names drawn receive the gold medals, the next two drawn have the silver medals, and the bronze medal goes to the remaining pupil.
How many different ways can the medals be allocated by this method?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
There are $6$ different students who could receive the first gold medal, then $5$ others for the second and $4$ remaining for the third. Therefore there are $6 \times 5 \times 4 = 120$ orders in which the medals can be presented. However, this counts each set of three people winning the medals in each of the six orders, so there are $120 \div 6 = 20$ sets of three people who could win gold.
For each of these, one of the remaining three people must win bronze and the others silver, so there are $20 \times 3 = 60$ ways in which the medals can be awarded.