Answer to Question #255356 in Discrete Mathematics for Solehah Halias

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. without loss of generality, we can assume the recorded results are n tuples(a₁,a₂,a₃,---,a_n)

each place can be occupied by 1 through 6 numbers.

the result of the first die is independent of second and soon, we can say each place has 6 choices , two places have6² , three places have 6³ , --- ,n places have 6ⁿ possibilities.

∴ the total number of available results is 6ⁿ.

b. if one place of this n tuple is occupied by 6, then there are n-1 places .

since the dies are indistinguishable, the place of its result is also indistinguishable in the n -tuple.

so, leave that place to 6 , the n-1 places can be occupied by the 6 available results in 6 ^n-1 results.

c. same as the case b. if two places are given to two 4's ,the remaining n-2 places can be occupied in 6 ^n-2ways.