## Thinking Mathematically (6th Edition)

Both involve combinations since order is unimportant. The one with the smaller number of combinations will be easier to win. (Your chance is better when you have one of 20 than when you have one of 200 possible combinations.) The 6/53 has ${}_{53}C_{6}=\displaystyle \frac{53!}{(53-6)!6!}=\frac{53\times 52\times 51\times 50\times 49\times 48}{1\times 2\times 3\times 4\times 5\times 6}$ $=$ 22,957,480 different combinations. The 5/36 has ${}_{36}C_{5}=\displaystyle \frac{36!}{(36-5)!5!}=\frac{36\times 35\times 34\times 33\times 32}{1\times 2\times 3\times 4\times 5}$ = 376,992 different combinations. The 5/36 lottery is easier to win.