求可计算正/负/零值N的Excel二项式系数(N,K)单公式

Hey there, I’ve got a single Excel formula that’ll handle binomial coefficients for positive, negative, or zero values of N (between -10 and 10) and K (0 to 10, per your range constraints) perfectly. It’ll even validate inputs and work for both single calculations and batch table use.

=IF(OR(K<0,K>10,N<-10,N>10),"Invalid Input",IF(K=0,1,IF(N>=0,FACT(N)/(FACT(N-K)*FACT(K)),PRODUCT(ROW(INDIRECT("1:"&K))+N-1)/FACT(K))))

How This Formula Works #

Let’s break down each part to make sense of it:

• Input Validation: The opening IF(OR(K<0,K>10,N<-10,N>10),"Invalid Input",...) checks if your N/K values fall outside the allowed range. If they do, it returns a clear error message instead of a wrong calculation.
• K=0 Edge Case: For any valid N, the binomial coefficient (\binom{N}{0}) is always 1. We handle this upfront to skip unnecessary math.
• Non-Negative N: When N is positive or zero, we use the standard factorial-based formula you mentioned: FACT(N)/(FACT(N-K)*FACT(K)). This is the classic method for calculating binomial coefficients with non-negative integers.
• Negative N: For negative N, we use the generalized binomial coefficient formula. The PRODUCT(ROW(INDIRECT("1:"&K))+N-1) computes the product of terms like (N \times (N+1) \times ... \times (N+K-1)), then dividing by (K!) (via FACT(K)) gives the correct generalized coefficient.

Batch Calculation Adaptation #

If you’re using this in a table (e.g., N in column A, K in column B starting at row 2), adjust the formula to reference your cells:

=IF(OR(B2<0,B2>10,A2<-10,A2>10),"Invalid Input",IF(B2=0,1,IF(A2>=0,FACT(A2)/(FACT(A2-B2)*FACT(B2)),PRODUCT(ROW(INDIRECT("1:"&B2))+A2-1)/FACT(B2))))

Drag this formula down the column, and it’ll automatically compute the binomial coefficient for every row of N/K pairs.

Example Test Values #

内容的提问来源于stack exchange，提问作者steveOw