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- Homework Statement:
- What is the total number of ways of choosing 5 non consecutive numbers from the set of first 15 natural numbers?

- Relevant Equations:
- Number of ways of choosing r out of n things is nCr

I have seen a solution for this question which was as follows,

first out of 15 elements, take away 5, thus there are 11 gaps created for the remaining 10 numbers (say N) as,

_N_N_N_N_N_N_N_N_N_N_

now, now we can insert back the 5 to comply with the non-consecutive stipulation

for which, number of ways=11C5 (answer)

But I can't understand how selection of 5 out of 11 gaps between 10 random numbers will ensure that the chosen numbers will be non-consecutive. If the question was finding the total number of ways for siting 5 boys between 10 girls such that no two boys sit together, then I could understand this solution but how does this method work here and how are these two situations similar?

first out of 15 elements, take away 5, thus there are 11 gaps created for the remaining 10 numbers (say N) as,

_N_N_N_N_N_N_N_N_N_N_

now, now we can insert back the 5 to comply with the non-consecutive stipulation

for which, number of ways=11C5 (answer)

But I can't understand how selection of 5 out of 11 gaps between 10 random numbers will ensure that the chosen numbers will be non-consecutive. If the question was finding the total number of ways for siting 5 boys between 10 girls such that no two boys sit together, then I could understand this solution but how does this method work here and how are these two situations similar?