Project Euler Forum

For the combinatorics problem 53, ²³C₁₀ is the same as ²³C₁₃. Do these count as two values or 1?

In other words, is the question asking for unique number of values or *total* number?

They count as two values.

The question is asking for *total* number.

Rereading it now, it seems ambiguous. Maybe it should read something like:

How many pairs of numbers (n,r), for 1<=n<=100, are there with nCr greater than one-million?

The phrase "values of nCr" in the original question probably shouldn't be used considering 23C10 = 23C13 are both counted but have the same value.

The phrase "How many values of ⁿC_r" can indeed be interpreted in two ways.
You can either consider the function (n,r) &rarr; ⁿC_r, which is the intended interpretation here, or you can consider the set of values and its cardinality.
Would "How many values of ⁿC_r, for 1 &le; n &le; 100, 0 &le; r &le; n, are greater than one-million?" be good?

daniel.is.fischer wrote: Would "How many values of ⁿC_r, for 1 &le; n &le; 100, 0 &le; r &le; n, are greater than one-million?" be good?

I think that is just as ambiguous. How about:
For 1 &le; n &le; 100, 0 &le; r &le; n, how often is ⁿC_r greater than one-million?

Or simply:
How many, not necessarily distinct, values of nCr, for 1<=n<=100, are greater than one-million?

As no further comments were given, I changed it to my version which is nearest to the original version.

Im having a ton of trouble with this one. My program will return 1144066 for ²³C₁₀, but my count is always wrong. tried changing it a little tons of times, really cant see anything wrong. anyone care to look at it? i wrote it in java.

If your code correctly returns the ⁿC_r function then it's probably miscounting. I suggest you re-check your code to make sure your're counting according to the problem rules, as I myself often get a wrong interpretation of those.

Next time remember to leave the problem link =)
Problem 53 (View Problem)

m34tcode wrote:Im having a ton of trouble with this one. My program will return 1144066 for ²³C₁₀, but my count is always wrong. tried changing it a little tons of times, really cant see anything wrong. anyone care to look at it? i wrote it in java.

If you don't manage to solve it with the above suggestion, I could take a look. You can send a PM.

Remember to only count the distinct values. For example ²³C₁₀ and ²³C₁₃ are both 1144066 but only count as one rather than two.
Edit: Ignore this post

jaap wrote:Remember to only count the distinct values. For example ²³C₁₀ and ²³C₁₃ are both 1144066 but only count as one rather than two.

This contradicts earlier posts.

hk wrote:

jaap wrote:Remember to only count the distinct values. For example ²³C₁₀ and ²³C₁₃ are both 1144066 but only count as one rather than two.

This contradicts earlier posts.

Sorry! I remembered it wrong and misunderstood the problem when I reread it. I even contradicted my previous own post (the thread was merged then).

Anyway, every (n,r) pair that gives ⁿC_r greater than a million should be included in the count.