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Match the pairs in the following questions by writing the corresponding letters only.

$\begin{array}{|cl|cl|} \hline A. & \text{The number of distinct binary tree with n nodes.} & P. & \frac{n!}{2} \\ \hline B. & \text{The number of binary strings of the length of 2n with an equal number of 0’s and 1’s} & Q. & \binom{3n}{n} \ \\ \hline C. & \text{The number of even permutation of n objects.} & R. & \binom{2n}{n} \\ \hline D. & \text {The number of binary strings of length 6n which are palindromes with 2n 0’s.} & S. & \frac{1}{1+n}\binom{2n}{n} \\ \hline \end{array}$

edited | 1.7k views
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The editing of this question has gone haywire. Some part of B and D is showing up in Q and S section. Same is the case in GO hardcopy.

1. - S Catalan number http://http://gatecse.in/wiki/Number_of_Binary_trees_possible_with_n_nodes
2. - R Choosing $n$ locations for $0$'s out of $2n$ locations. The remaining $n$ locations are filled with $1$'s (no selection required).
3. - P An even permutation is a permutation obtainable from an even number of two-element swaps, For a set of $n$ elements and $n>2$, there are $n!/2$ even permutations.
Ref - http://mathworld.wolfram.com/EvenPermutation.html
4.  - Q

Length $= 6n$, as it is palindrome, we need to select only the first half part of the string.

Total length to consider is $3n$ (Remaining $3n$ will be revese of this $3n$)

Now, choose $n \ 0's$ out of $3n$. So Q is correct for D.

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Explain even permutations with an example
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What is even permutation ???I'm not getting from above link ??
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The option for the number of binary trees is incorrect.

Number of BSTs (Binary Search Trees) = nth Catalan Number, whereas number of Binary Tress = (n!) * nth Catalan number.

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then what will be the value of odd permutations?
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same as even permutations, $\frac{n!}{2}$

A - S
B - R
C - P
D - Q

It will be $\frac {n!}{2}$

using the property of symmetry.

http://mathworld.wolfram.com/EvenPermutation.html

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An even permutation is a permutation obtainable from an even number of two-element swaps, For initial set 1,2,3,4, the twelve even permutations are those with zero swaps: (1,2,3,4); and those with two swaps: (1,3,4,2, 1,4,2,3, 2,1,4,3, 2,3,1,4, 2,4,3,1, 3,1,2,4, 3,2,4,1, 3,4,1,2, 4,1,3,2, 4,2,1,3, 4,3,2,1). etc.

For a set of  n elements and n>2, there are n! / 2 even permutations, which is the same as the number of odd permutations

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