How many different meals can be ordered?

Question

Ron must choose one entree, 3 vegetables, and one dessert. There are 5 entrees, 10 vegetables, and 3 desserts to choose from. How many different meals can Ron order?

Answer 1

There are 5 choices for the one entree, and 3 choices for the one dessert.  So there are 15 total possibilities for these 2 parts of the meal.  

 

Figuring out the number of vegetable options is a little more complicated.  For the 1st vegetable there are 10 choices, for the 2nd there are 9 choices, and for the 3rd there are 8 choices.  So it would appear that there are 10*9*8 vegetable choices.  But things are not quite as simple as that.  If you pick peas, corn and carrots for your vegetables you will get the same choices as if you picked corn, carrots and peas - just in a different order.  The order that you pick the vegetables in doesn't matter, and any set of 3 vegetables can be selected in 3*2*1 = 6 different ways (there are 3 choices for the 1st vegetable in the list, 2 choices for the 2nd vegetable in the list, and 1 choice for the last vegetable in the list).  So we must divide 10*9*8 by 3*2*1 to get 5*3*8 = 120 vegetable choices.  

 

So the total number of different meals is 15*120 = 1800  

 

By the way, there is a special name and formula for the method of selecting k objects from n choices like we did for the vegetables.  It is called 

 

_nC_k  which is read "n choose k" and _nC_k = n!/(k!(n-k)!)  

 

So ₁₀C₃ = 10!/(3!7!) = 10*9*8/6= 120