# Permutation/combination problems

1. Alice rolls an ordinary six-sided die 8 times in succession, and the numbers are recorded (in order). How many possible outcome sequences contain exactly two occurrences of "6" Briefly explain your answer.

2.Bob tosses a coin 20 times and gets 13 heads and 7 tails. In how many ways can these tosses result in exactly three (non-empty) blocks of consecutive heads? For example, HHHTTHHHHHHHHHTTTHTT has three blocks of consecutive heads. Show all steps of your solution. (Hint: Think of distributing apples and oranges to children such that each child gets at least one apple.)

any help is appreciated. thanks in advance

HallsofIvy
Homework Helper
Well, the problem itself gives a hint! Show us what you have tried on this so we will know where you need help. Do you know anything about "permutations and combination"? What formulas do you have to use?

there is no specific formula that I need to use..
For question 2, I think im able to use this formula though (n+r-1)/r
I have no idea how to apply this formula into that question.

As an example, if there are 7 bananas, 6oranges, distributed to 4 children, such that each child gets at least 1 banana. in how many ways?

ans: (4+3-1)/3 * (4+6-1)/6

cristo
Staff Emeritus
1. How many possible outcome sequences are there if out of eight throws, only one six is thrown?

Now, each of these sequences has one six "fixed" in position. How many ways are there to arrange another six in the remaining slots.

Now you have answered the above questions, can you answer the problem?

Do you know anything about permutations/combinations? Do you know a formula to work out such questions?

if one six is thrown 8 times, there will be 6*6*6*6*6*6*6*6, but I do not know how to get 2 consecutive of 6's.

HallsofIvy