combination

Question

How many binary sequences of length 10 are possible with exactly 4 zeros and no two zeros are consecutive?

Answer 1

Its given exactly 4 zero it means we have 6 1's  which can be arranged in any order therefore 1! (since all 1 are identical)

1 1 1 1 1 1      since no two 0 should be consecutive we have total 7 space between each 1 we can place one 0

 so select 7 places for 4 0's(all 0 are identical so we use the combination rather than permutation for arrangement) = 7C4

                   Therefore total binary string = 1! *7C4 =35

Comments

Why isn't  7 P 4 here.Plz explain @Umang

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you can arrange identical element only in one way. i.e. why we only need to select the empty places.

Answer 2

Now, given that there are 4 zeros and they arent consecutive.

So, minimal situation is 

_ 0 1 0 1 0 1 0 _

Once arranged this way, we can place remaining 3 zeros each in 5 ways as shown:

_ 0 _ 0 _ 0 _ 0 _

So, answer = 5³ = 125