מחשבון תמורות – P(n,r) וצירופים C(n,r)

איך להשתמש במחשבון זה

1. הזן n (total items)
2. הזן r (items chosen)
3. הזן Calculate
4. לחץ על כפתור חשב
5. קרא את התוצאה המוצגת מתחת למחשבון

Permutations vs Combinations: The Key Difference

Permutations P(n,r): Count ordered arrangements. Order matters. "How many ways can you arrange 3 people from a group of 10?" ABC ≠ BAC ≠ CAB — these are different permutations.

Combinations C(n,r): Count unordered selections. Order doesn't matter. "How many ways can you choose 3 people from a group of 10?" {A,B,C} = {B,A,C} = {C,A,B} — all the same combination.

The formulas:

• P(n,r) = n! / (n−r)!
• C(n,r) = n! / (r! × (n−r)!)

C(n,r) = P(n,r) / r! — combinations equal permutations divided by the number of ways to arrange r items (since order doesn't matter in combinations).

How to Calculate P(n,r): Step by Step

P(n,r) counts ordered arrangements of r items chosen from n items:

P(n,r) = n × (n−1) × (n−2) × ... × (n−r+1) = n! / (n−r)!

Example: P(10, 3)

1. P(10,3) = 10! / (10−3)! = 10! / 7!
2. = (10 × 9 × 8 × 7!) / 7!
3. = 10 × 9 × 8 = 720

There are 720 ways to arrange 3 people from a group of 10 in order (e.g., 1st, 2nd, 3rd place finishers in a race).

How to Calculate C(n,r): Step by Step

C(n,r) = n! / (r! × (n−r)!). Also written as ⁿCᵣ or "n choose r" or (n r).

Example: C(10, 3)

1. C(10,3) = 10! / (3! × 7!)
2. = (10 × 9 × 8 × 7!) / (6 × 7!)
3. = (10 × 9 × 8) / (3 × 2 × 1)
4. = 720 / 6 = 120

There are 120 ways to choose 3 people from a group of 10 (e.g., selecting a committee of 3 from 10 candidates).

Factorials and Large Numbers

Factorials grow extremely rapidly: 10! = 3,628,800; 20! ≈ 2.43 × 10¹⁸; 52! ≈ 8.07 × 10⁶⁷ (number of ways to shuffle a deck of cards).

Computing large factorials directly overflows standard calculators. The formula P(n,r) = n!/(n-r)! can be simplified by canceling common terms in the numerator and denominator rather than computing raw factorials.

For very large n and r, Stirling's approximation (n! ≈ √(2πn)(n/e)ⁿ) or logarithms of factorials are used.

Real-World Examples

When to use permutations:

• Race finishing positions (1st, 2nd, 3rd out of 20 runners)
• Passwords and PINs (4-digit PIN from 0–9)
• Arranging books on a shelf
• Assigning roles to people

When to use combinations:

• Selecting a team/committee (order of selection doesn't matter)
• Lottery numbers (choosing 6 from 49)
• Choosing toppings for a pizza
• Selecting questions from an exam to answer

Lottery example: UK National Lottery: C(59,6) = 45,057,474. Your odds of winning the jackpot are approximately 1 in 45 million.

Pascal's Triangle and Combinations

Pascal's Triangle is a triangular array where each number is the sum of the two above it. Each entry is a combination: row n, position r = C(n,r).

Row 0: 1 = C(0,0)
Row 1: 1 1 = C(1,0) C(1,1)
Row 2: 1 2 1 = C(2,0) C(2,1) C(2,2)
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1

Pascal's Triangle enables rapid combination calculation and appears in the binomial theorem: (a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ.

Special Cases and Identities

• C(n,0) = C(n,n) = 1: One way to choose none or all items
• C(n,1) = n: n ways to choose 1 from n
• C(n,r) = C(n, n−r): Symmetry — choosing r is equivalent to choosing which n−r to leave out
• P(n,1) = n: n ways to choose 1 in order
• P(n,n) = n!: All arrangements of n items
• P(n,0) = C(n,0) = 1: One way to arrange/choose nothing