GATE CSE 2014 Set 1 | Question: 49

Question

A pennant is a sequence of numbers, each number being $1$ or $2$. An $n-$pennant is a sequence of numbers with sum equal to $n$. For example, $(1,1,2)$ is a $4-$pennant. The set of all possible $1-$pennants is ${(1)}$, the set of all possible $2-$pennants is ${(2), (1,1)}$ and the set of all $3-$pennants is ${(2,1), (1,1,1), (1,2)}$. Note that the pennant $(1,2)$ is not the same as the pennant $(2,1)$. The number of $10-$pennants is________

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Answer 1

Let us take the example of 4 pennant

If we have to make it with :-

4 positions number of 2's woulld be 0

3 positions number of 2's woulld be 1

2 positions number of 2's woulld be 2

1 positions number of 2's woulld be 3-> which is just not possible as 3 2's need 3 positions ovviosly

so for 4 pennants its 4C0 +3 C1 +2C2  = 5

similarly for 10 pennant its

10C0 + 9C1 + 8C2 + 7C3 + 6C4 + 5C5 =89

Answer 2

This is the best way to do this

I hope that its the easiest possible explanation to this question