How do you prove nCr=n! /r! (n-r)! With clear steps?

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The nCr formula, also known as the combination formula, calculates the number of ways to choose r items from n items where order does not matter. This fundamental concept is essential in combinatorics, probability theory, and statistical analysis.

Formulas Table

Formula Type	Mathematical Expression	Alternative Notation	Explanation
Basic nCr Formula	nCr = n! / (r! × (n-r)!)	C(n,r) = n! / (r! × (n-r)!)	Number of combinations of n items taken r at a time
nPr Formula	nPr = n! / (n-r)!	P(n,r) = n! / (n-r)!	Number of permutations of n items taken r at a time
Relationship Formula	nCr = nPr / r!	C(n,r) = P(n,r) / r!	Combinations equals permutations divided by r factorial
Factorial Definition	n! = n × (n-1) × (n-2) × ... × 2 × 1	0! = 1 (by definition)	Product of all positive integers up to n

Essential Properties and Special Cases

Property/Case	Formula	Explanation	Example
Symmetry Property	nCr = nC(n-r)	Choosing r items equals choosing (n-r) items to exclude	5C2 = 5C3 = 10
Boundary Cases	nC0 = nCn = 1	Only one way to choose nothing or everything	7C0 = 7C7 = 1
Single Selection	nC1 = n	n ways to choose one item from n items	8C1 = 8
Pascal's Identity	nCr = (n-1)Cr + (n-1)C(r-1)	Used to construct Pascal's triangle	5C2 = 4C2 + 4C1
Sum Property	Σ(nCr) = 2ⁿ (r=0 to n)	Sum of all combinations equals 2 to the power n	3C0 + 3C1 + 3C2 + 3C3 = 8 = 2³

Probability Applications

Application	Formula	Use Case	Example
Basic Probability	P(Event) = nCr / Total Combinations	Calculating probability using combinations	P(2 heads in 4 flips) = 4C2 / 2⁴
Binomial Probability	P(X = k) = nCk × p^k × (1-p)^(n-k)	Probability of exactly k successes in n trials	P(exactly 3 successes in 10 trials)
Hypergeometric	P(X = k) = (KCk × (N-K)C(n-k)) / NCn	Sampling without replacement	Drawing cards from a deck

Advanced Relationships

Relationship	Formula	Mathematical Context
Binomial Theorem	(a + b)ⁿ = Σ(nCr × aⁿ⁻ʳ × bʳ)	Expansion of binomial expressions
Vandermonde's Identity	(m+n)Cr = Σ(mCk × nC(r-k))	Sum over all valid k values
Chu-Vandermonde	Σ(mCr × nCs) = (m+n)C(r+s)	When summing over specific ranges

Computational Formulas for Large Numbers

Method	Formula	Advantage
Multiplicative Form	nCr = (n × (n-1) × ... × (n-r+1)) / (r × (r-1) × ... × 1)	Avoids large factorial calculations
Recursive Formula	nCr = (n × (n-1)C(r-1)) / r	Efficient for sequential calculations
Logarithmic Form	log(nCr) = log(n!) - log(r!) - log((n-r)!)	For very large numbers to prevent overflow

Mistakes to Avoid

Error Type	Incorrect	Correct	Note
Order Confusion	Using nPr for unordered selection	Use nCr for combinations	Order doesn't matter in combinations
Zero Factorial	0! = 0	0! = 1	By mathematical convention
Negative Values	nCr where r > n	nCr = 0 when r > n	Cannot choose more than available
Non-integer Values	Using decimals for n or r	Only use non-negative integers	Combinations require whole numbers

How do you prove nCr=n! /r! (n-r)! With clear steps?

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