Number of Solutions

A famous mathematical combinatorics puzzle that is a must know

How many distinct solutions does the following equation have?

a+b+c+d = 100

where a ∈ {1,2,3,...}, b ∈ { 2,3,4,...}, c ∈ {0,1,2,3,...} , d ∈ {0,1,2,3,...}

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Try to solve this problem yourself before moving on to the solution below

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Solution

This is the famous coin and stick problem with a given constraint. Consider there are 100 identical coins and 3 identical sticks. Given a random permutation, we count the number of coins till the first stick & give it to a, count the number of data points between first and second stick & give them to b and so on. Take some time to convince yourself that these two problems are equivalent.

This time we have placed restrictions on the domains of the variables. Due to the restrictions placed on a and b, we have already assigned a minimum of (1 + 2 = 3) coins before the first stick and between the first and second stick. We have 97 coins left to assign along with 3 dividers. Therefore, the total number of items to assign amounts to 97 + 3 = 100.

Therefore the total number of unique permutations are n choose m ( i.e 100 choose 3 ) = 161700

\({100 \choose 3} = 161700\)

😃 Over to you: Were you able to solve this?

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