I have recently been reading Sedgewick's Algorithms Book, and I came across an example I didn't quite understand...
Using this code, Sedgewick mentions that the if statement is run precisely N x (N-1)(N-2)/6 times.
I don't quite understand why that is.... I see that there are the triple nested loops, each with an upper bound that is increasing, but why is the value divided by six?
I ask for your understanding that I am not good at math, so the explanation may be a little bit cheesy.
In order to 'if(a[i] + j[i] + k[i]) statement' inside the code be executed The variables i, j, and k that point out an array 'a' index in Tripple for loop must fit the condition of the statement (0<= i <j<k <N), right?
For example, N is 4.
It seems to be available through the permutation (4P3 = 4^3) that selects three of the four, but if 4 is selected in 'i' as shown in the picture above, you can see that progress is blocked in 'j' for-loop (j = 4+1; j<4; j++).
So We can't get the number of times an if statement will run with a simple sequence.
What we need is a combination (nCr).
When index flows from 0 to N-1,
The number of cases (i,j,k) that satisfy i < j < k(=> if statement can run) can be obtained by the formula nC3.
According to this formula, if statement is run precisely N x (N-1)(N-2)/6 times
I hope you understand it well and if you don't, please leave a comment! Have a nice day!