Question

How many paths from (0 0) to (n n) without crossing the diagonal?

Answer 1

Answer:

Pn

Step-by-step explanation:

There are Pm ways to go to the point C(m,m). We know that for the rest part of the path we wont touch the diagonal anymore, so that means that our path lies under the line DE. Which means, that for the rest part of the path we have Pn−(m+1) ways to do that. So the group of C(m,m) consists of Pm⋅Pn−(m+1) good paths. We may take m=0,…,n−1 and then the sum of all possible paths is ∑n−1m=0Pm⋅Pn−m−1=Pn

Answer 2

many paths from (0,0) to (n,n) without crossing the diagonal

Catalan numbers

   It is a numbers are a sequence of positive integers that appears in many counting problems in combinatorics. They count certain types of lattice paths, permutations, binary trees, and many other combinatoric objects.

    They satisfy a fundamental recurrence relation, and have a closed-form formula in terms of binomial coefficients.

       $C_{n}=\frac{1}{n+1}\left(\begin{array}{c}{2 n} \\{n}\end{array}\right)$

Given Below shown the example

  make a triangle with all the points from (0,0) to (5,5) below the diagonal drawn on it.

Refer the answer in the picture below.

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• brainly.in/question/7234292