Factorials play a fundamental role in mathematics, with applications ranging from permutations and combinations to number theory and binomial expansion. This article dives deep into the concept of factorials, their applications, and problem-solving techniques.
The factorial of a non-negative integer nn, denoted by n!n!, is the product of all positive integers less than or equal to nn.
Visual Suggestion:
An infographic illustrating how 5!5! is calculated step by step.
The highest power of a prime pp in n!n! is calculated using the formula:
Power of p=⌊n/p⌋+⌊n/p2⌋+⌊n/p3⌋+…\text{Power of } p = \lfloor n/p \rfloor + \lfloor n/p^2 \rfloor + \lfloor n/p^3 \rfloor + \dots
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A flowchart showing steps to calculate trailing zeros in a factorial.
Factorials are key in permutations and combinations:
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A table comparing permutations and combinations with examples.
Factorials determine coefficients in binomial expansion:
(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k
Here, (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}.
Visual Suggestion:
A diagram showing how factorials contribute to the binomial expansion formula.
Problem: A number n!n! is written in base 6 and base 8. Its base 6 representation ends with 10 zeros, and base 8 ends with 7 zeros. Find the smallest nn.
Solution:
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Side-by-side base 6 and base 8 calculations.
Problem: How many ways can 8 people line up for a photo?
Solution:
8!=40,3208! = 40,320
Visual Suggestion:
An image showing people lined up with labels for each position.
Problem: How many 5-digit numbers can be formed using the digits 1, 2, 5, 7, and 8 without repetition?
Solution:
5!=1205! = 120
Problem: Evaluate 5!×(6−3)!5! \times (6-3)!.
Solution:
5!×3!=120×6=7205! \times 3! = 120 \times 6 = 720
Factorials are a cornerstone of mathematics, with applications ranging from simple arrangements to complex algebraic expansions. Mastering factorials can unlock solutions to many combinatorial and number theory problems.