Binomial Coefficient Problem Send Feedback A Ninja was learning how to calculate the binomial coefficient. But, learning that alone won’t help the ninja since a lot of problems are required to be solved as homework. Since the ninja is old-fashioned and doesn’t know that a computer can do the same homework in a matter of a few seconds you will have to help with the problems. You need to complete a function for the ninja that can calculate the binomial of a number where when given two integers 'N' and 'R', the program can calculate its respective binomial coefficient nCr. Since the answer may be very large, calculate the answer modulo 10^9 + 7. For Example: Input: 'N' = 5, 'R' = 2 Output: 10 The value of 5C2 is 10 Input Format N R space separated in one line Output Format
Question
Binomial Coefficient Problem Send Feedback A Ninja was learning how to calculate the binomial coefficient. But, learning that alone won’t help the ninja since a lot of problems are required to be solved as homework. Since the ninja is old-fashioned and doesn’t know that a computer can do the same homework in a matter of a few seconds you will have to help with the problems. You need to complete a function for the ninja that can calculate the binomial of a number where when given two integers 'N' and 'R', the program can calculate its respective binomial coefficient nCr. Since the answer may be very large, calculate the answer modulo 10^9 + 7. For Example: Input: 'N' = 5, 'R' = 2 Output: 10
The value of 5C2 is 10 Input Format N R space separated in one line Output Format
Solution
Sure, here is a Python function that calculates the binomial coefficient:
def binomial_coefficient(n, r):
MOD = 10**9 + 7
if r > n:
return 0
if r > n - r:
r = n - r
num = 1
den = 1
for i in range(r):
num = (num * (n - i)) % MOD
den = (den * (i + 1)) % MOD
return (num * pow(den, MOD - 2, MOD)) % MOD
This function uses the formula for the binomial coefficient, which is nCr = n! / (r!(n-r)!). However, since the factorial of a number can be very large, we calculate the numerator and denominator separately and use the modulo operation to keep the numbers manageable. We also use the property of modular arithmetic that (a/b) mod p = (a * (b^(-1))) mod p, where b^(-1) is the multiplicative inverse of b under modulo p. This is calculated using the pow function with the third argument as MOD - 2, based on Fermat's Little Theorem.
Similar Questions
List two ways to find binomial coefficients. (Select all that apply.)Binomial TheoremPythagorean TheoremSum of CoefficientsPascal's TriangleEuler's MethodSum of Powers
You are given two integers 'n' and 'r' and a prime number 'p'. Your task is to find (nCr) % p where nCr can be calculated as n! / (r! * (n - r)!).Note :N! = 1 * 2 * 3 *... NInput format :The first line of input contains a single integer 'T', representing the number of test cases. The first line of each test case contains three space-separated integers 'n', 'r', and 'p'.Output format :For each test case, print a single line containing an integer representing the value of (nCr) % p.The output of each test case will be printed on a separate line.Note:You don't need to input or print anything, it has already been taken care of. Just implement the given function.Constraints :1 <= T <= 5 1 <= n, r, p <= 5 * 10 ^ 2p is prime number.Time limit: 1 sec.Sample Input 1 :2 5 2 114 3 13Sample Output 1 :104Explanation for Sample Output 1:In test case 1, n = 5, r = 2, and p = 11n C r = 5 C 2 = (5 * 4) / (2!) = 10n C r % p = 10 % 11 = 10. So the answer will be 10.In test case 2,n = 4, r = 3, and p = 13 n C r = 4 C 3 = 4 C 1 = 4 n C r % p = 4 % 13 = 4. So the answer will be 4.Sample Input 2 :25 2 1710 2 13Sample Output 2 :106
According to the Binomial Theorem,(a + b)n = an + nC1an − 1b + ... + nCran − rbr + ... + nCn − 1abn − 1 + bnthe given expression is already in the form (a + b)n, where a = x, b = and n = .
Given two numbers N and R, find the value of NCR.Input FormatThe first and only line of input contains integers N and R.Output FormatPrint the value of NCRConstraints1 <= N <= 101 <= R <= 10ExampleInput5 3Output10
Find the specified nth term in the expansion of the binomial.(x – 10z)7, n = 6Step 1According to the Binomial Theorem,(a + b)n = an + nC1an − 1b + ... + nCran − rbr + ... + nCn − 1abn − 1 + bnthe given expression is already in the form (a + b)n, where a = x, b = $$−10z and n = 6 6 .Step 2Now use the Binomial Theorem to find the 6th term in the expansion of ((x) + (−10z))7. Note that the rth term is nCr − 1an − (r − 1)br − 1 and not nCran − rbr. Also recall that nCr = n!r!(n − r)!, and that (ab)n = an bn.Therefore, the 6th term in the expansion of (x − 10z)7 is as follows.7C5x7 − 5(−10z)5= 7!5! !x2(−10z)5 = 7 · 6 · 5!5! 2!x2(−10)5(z)5 = 7 · 62!x2(−100,000z5) = 7 · 62 · 1x2(−100,000z5) = x2(−100,000z5) =
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