Module 3 · Number System
Chapter 39: Factorials
Number System
3 topics
What this chapter covers
The 3 topics below make up Factorials on the CAT. Click any topic to open its notes, examples and practice test.
Study material & tests
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📝 Chapter test
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Formula Reference Sheet
This chapter
Core definitions & identities
| Factorial | n! = 1 × 2 × 3 × … × n |
|---|---|
| Empty product | 0! = 1! = 1 |
| Recurrence | n! = n × (n − 1)! |
| Ratio of factorials | n! / (n − r)! = n(n−1)…(n−r+1) |
| Combination | ⁿCᵣ = n! / [r!(n − r)!] |
Power & zero-counting tools
| Highest power of prime p in n! (Legendre) | ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … |
|---|---|
| Trailing zeros of n! | ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + … |
| Power of composite (e.g. 6 = 2×3) | min of the prime-factor powers in n! |
| Power of pᵏ (e.g. 9 = 3²) | ⌊ (power of p in n!) / k ⌋ |
| Quick count of multiples of m up to n | ⌊n/m⌋ |
CAT reference
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