is there any method to count the number of ways in which 100 can be expressed as sum of even integers?

Question

can anyone help me out with this problem, which is as follows; my friends gave me the challenge to count the no. of ways in which 100 can be expressed as the sum of even positive integers. I spent hours and got an answer, and I am a bit unsure that the answer is approximately 589826. can anyone help me out, please? I am actually searching for an answer, and I am trying to ask that how many ways are there to write 100 as the sum of positive even integers.

Answer 1

EDIT: actually, on second thought, the answer depends. Do you consider, for example, ${2+4}$ the same as ${4+2}$? If you do, then you can follow what @user2661923 said in the comments. Otherwise, you must do the following: note as stated by @Ritam_Dasgupta, your problem will be in one-to-one correspondence to finding partitions of ${50}$. Given a partition of ${50}$, $$ 50 = k_1 + ... + k_n $$ then you get $$ 100 = 2k_1 + ... + 2k_n $$ i.e. you get ${100}$ as the sum of even positive integers. Likewise for $$ 100 = 2c_1 + ... + 2c_n $$ then dividing both sides by $2$ gives a partition of ${50}$. It's easy to see with a little thought these will be in one-to-one correspondence.

Solving partitions is a highly non-trivial problem, you can approximate the answer by using: $$ p(n)\sim \frac{1}{4n\sqrt{3}}\exp\left(\pi \sqrt{\frac{2n}{3}}\right) $$ plugging in ${50}$ will give you a number much larger than yours stated.