Question

A TV show has 26 episodes. Due to the lack of time, you can onlywatch 5 of them. However, in order to get a gist of what the showis about, you do not want to watch any two consecutive episodes.How many selections of 5 episodes are there with no two consecutiveepisodes chosen? Assuming that the order of selection is irrelevant.(Hint:this is a Stars and Bars problem)

Answer 1

Explanation:

Recall that: The number of combination to select any k consecutive element from n consecutive term is given by the equation
`^(n-k+1)C_k`

`= \begin {pmatrix} ^(n-k+1)_(k) \end {pmatrix}`

where:

n = 26

k = 5

`^(n-k+1)C_k= \begin {pmatrix} ^(26-5+1)_(5) \end {pmatrix}`

`^(n-k+1)C_k= \begin {pmatrix} ^(22)_(5) \end {pmatrix}`

`^(n-k+1)C_k= (22!)/(5!(22-5)!)`

`^(n-k+1)C_k= (22!)/(5!(17)!)`

`\mathbf{^(n-k+1)C_k=26334}`

Therefore, there are 26334 selection of 5 episodes with no two consecutive episodes chosen