Question

If n fair six-sided dice are tossed and the numbers showing on top are recorded, how many

(a) record sequences are possible?

(b) sequence contain exactly one six?

(c) sequences contain exactly four twos, assuming n >= 4 

Answer 1

(a) For each cube there are 6 possible outcomes. Then for n cubes there will be $6^{n}$ possible sequences

(b) We have exactly one six in n tosses. Since number six can occur anywhere in the sequence(but exactly once), there is n such places( 6 on the first place, on the second, ... , on the nth place). On each of the others (n-1) places can occur any number except of 6. So, there are totally $n*5^{n-1}$ such sequences

(c) sequences contain exactly four twos, assuming n ⩾ 4. There is ${n \choose 4}$ places where twos can be placed. On the others (n-4) places can occur anything except of 2. So, there is $5^{n-4}$ possible sequences for any of the ${n \choose 4}$ places where 2 occured. Totally we got ${n \choose 4} *5^{n-4}$ sequences