If n fairsix-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?

(a). Without loss of generality, we can assume the recorded results are n tuples $(a1,a2,a3,......,an)$

each place can be occupied by 1 through 6 numbers.

the result of the first die is independent of the second and so on, we can say each place has 6 choices, two places have 62 , three places have 63 , --- , n places have 6n possibilities.

∴ the total number of available results is 6n.

(b). n six-sided dice

Sequence contains only one 6 = $^nC_1(5)^{n-1}= n(5)^{n-1}$

(Since from n place select one place 6 there and remaining (n-1) place any 5 number could be placed)

(c). Sequence contains exactly four 2's $=\ ^nC_4(5)^{n-4}$

(Select 4 places from n and place 2 there and remaining (n-4) place any 5 number could be placed )