Query on a topic on Permutation & Combination concepts by Gaurav Sharma

Permutation & Combination concepts by Gaurav Sharma  Part (5/5)
For the question The number of integral even sided triangle with perimeter 180
max side could be 89 but it was reduced to 88 as sides are of even magnitude.
But in next question, Find the number of triangle having odd integral sides of perimeter equal to 153 max value of a side could be 76, but due to sides being odd, it should have been reduced to 75, but the max value was taken as 76 only. Why this difference? Can someone please explain.

First question, where sides are even, we got max side as 89 ( = 180/2  1).
As we need all sides even, we represented the possible sides as 89  (2a + 1) etc. (odd  odd = even). This is same as 88  2a. Which is how it is represented in the solution for the first question.For the second question sides are represented as 76  (2a + 1) (even  odd = odd)
If you want to adjust for the odd criteria in the maximum value (like in the first question), it will become 75  2a (odd  even = odd) which is nothing but 76  (2a + 1).So this is the same thing done in two different ways. You can choose the method you liked and apply the same in both problems and still get the same answer.
Happy Learning!

@zabeer Thanks for your reply, I was eagerly awaiting for the same.

It means if the question was, => Find the number of triangle having odd integral sides of perimeter equal to 151, then the max value for a side would have been 75, and so if we wish to use 2a1, then we would have used 74 instead of 75. OR we would have used 75 only if the variable substitution we wish to use is 2a. Am I Correct ?

@test1234 that's my understanding too. You can cross check with the formula mentioned in the article.

@zabeer thanks again