# High School Mathematics Extensions/Supplementary/Basic Counting/Solutions

## Solutions to exercises

edit### Exercise 1

editThere would be *4!* ways if the letters were all distinct. Since "O" is repeated twice (*2!* permutations), every different arrangement is counted *2!* times; therefore there are

ways.

### Exercise 2

editThere are thirteen diamonds in the deck; hence, there are

ways.

### Exercise 3

editSince the lettuce is always on top, we just consider all the possible arrangements of *ham - cheese - tomato - salami* such that the ham is never next to the salami.
We count the permutations of the four elements,

and then deduct the permutations in which the ham and the salami are together. That is, the permutations of *(ham & salami) - cheese - tomato* and *(salami & ham) - cheese - tomato*:

Joey can arrange the ingredients in

different ways.

### Exercise 4

editWe count the number of ways of choosing three flavours out of ten and discount the number of ways of choosing one flavour, chocolate, and vanilla, that is the number of ways of choosing one flavour out of the eight left:

Rachel can order *112* different ice cream cups.

### Exercise 5

editWe count the number of combinations of one, two, three, four, and five ingredients out of five available. That is the sum of five terms *5 choose k*, where *k* equals *1, 2, 3, 4, 5*:

Then we subtract to each term the number of combinations of ham, salami and *k - 2* ingredients out of the 3 left (note that *5 choose 1* does not need any deduction):

Joey can make

different sandwiches.