Question

If a die is rolled 25 times, there are 6²⁵ different sequences possible. The following question asks how many of these sequences satisfy certain conditions. HINT [Use the decision algorithm discussed in Example 3 of Section 6.3.]

What fraction of these sequences have exactly five 1s? (Round your answer to four decimal places.)

ANS is not 0.0105

Answer 1

Answer: If a die is rolled 25 times, there are 6²⁵ different sequences possible. The following question asks how many of these sequences satisfy certain conditions.

What fraction of these sequences have exactly five 1s?

Solution:

We have rolled die 25 times.In those 25 rolls we want there to be exactly five 1's.

Total possible sequence is 6^25

Since, there are five places have exactly 1's = 25-5 = 20

Number of possible sequence 20 other numbers =5^20

Number of ways by which exactly five 1s placed in 25 places is 25C5

Total number of possible sequence of exactly five 1s =

25C5 * 5^20

Therefore, fraction of sequences have exactly five 1s

= 25C5 * 5^20 / 6^25

= 0.1782

Therefore, the fraction of these sequences have exactly five 1s = 0.1782.