How to calculate 200C65 (for binomial distribution formula)

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SUMMARY

The discussion focuses on calculating the binomial coefficient 200C65, which presents challenges due to the large number size. Users attempted to compute this using calculators but encountered errors. A brute force solution is suggested, involving the multiplication of fractions from (200/65) down to (136/1). Additionally, Stirling's approximation is recommended for estimating large factorials, providing a method that is accurate to about 1% for 65!. However, for exact calculations, users are advised to consider the limitations of their calculators and utilize upper and lower bound formulas for verification.

PREREQUISITES
  • Understanding of binomial coefficients and their notation (nCr)
  • Familiarity with factorial calculations and their properties
  • Knowledge of Stirling's approximation for large factorials
  • Basic proficiency in using programmable calculators or computational tools
NEXT STEPS
  • Research the implementation of Stirling's approximation in programming languages like Python
  • Explore the use of combinatorial functions in libraries such as SciPy for Python
  • Learn about upper and lower bound techniques for verifying large combinatorial calculations
  • Investigate the limitations of various calculators when handling large integers
USEFUL FOR

Mathematicians, statisticians, students studying combinatorics, and anyone needing to compute large binomial coefficients accurately.

socrates_1
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Hi, I have tried to calculate 200C65 on my calculator but the calculator gives an error. Do u know how to do it?
I also tried to calculate it through the formula with the ! but doesn't give an answer.
 
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socrates_1 said:
Hi, I have tried to calculate 200C65 on my calculator but the calculator gives an error. Do u know how to do it?
I also tried to calculate it through the formula with the ! but doesn't give an answer.

I don't know the limit on number size for your calculator, but the simplest brute force solution would be:
(200/65)(199/64)...(136/1)
 
Stirling's formula is a way of approximating large factorials.

Mathman's approach is easy and exact if you have a computer or programmable calculator, but error prone if you are doing it manually. Stirling's approximation is (according to Wikipedia) accurate to about 1% for 65!. [strike]That's fine for your application, since you know that 200!/135!65! must be an integer - just use the approximation, do the division, and round off.[/strike]

Edit: Sorry - it's been a long day. 1% of 200! is a big number; my approach will not do if you need an exact answer. However, it'll tell you right off if your calculator can handle the answer. Also, you can use the upper and lower bound formulae to get a sanity check on your answer.

Off to bed now...
 
Last edited:

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