Question

12!/(9!*3!)=? 12!/(10!*3!)=? 12!/(11!*3!)=?

Answer 1

Факториал числа n определяется:  $n!=1\cdot2\cdot3\cdot...\cdot n$

$1)~\displaystyle \frac{12!}{9!3!}= \frac{9!\cdot10\cdot11\cdot 12}{9!3!}= \frac{10\cdot11\cdot12}{1\cdot2\cdot3}= 220$

$2)~\displaystyle \frac{12!}{10!3!}= \frac{10!\cdot 11\cdot12}{10!\cdot1\cdot 2\cdot 3}= \frac{11\cdot12}{6} =22$

$3)~ \displaystyle \frac{12!}{11!3!}= \frac{11!\cdot12}{11!3!}= \frac{12}{1\cdot2\cdot3}=2$

Answer 2

См скриншот
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Answer 3

To simplify these expressions, let's first understand what factorials mean. The factorial of a positive integer n (denoted as n!) is the product of all positive integers from 1 to n.

For example:

• 5! = 5 × 4 × 3 × 2 × 1 = 120
• 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Now, let's calculate the given expressions:

1. 12! / (9! * 3!)
   • 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
   • 9! = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
   • 3! = 3 × 2 × 1

Now, we can cancel out the common terms in the denominator with the numerator:

12! / (9! * 3!) = (12 × 11 × 10) / 1 = 1320

2. 12! / (10! * 3!)
   • 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
   • 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
   • 3! = 3 × 2 × 1

Now, we can cancel out the common terms in the denominator with the numerator:

12! / (10! * 3!) = (12 × 11) / 1 = 132

3. 12! / (11! * 3!)
   • 12! = 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
   • 11! = 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
   • 3! = 3 × 2 × 1

Now, we can cancel out the common terms in the denominator with the numerator:

12! / (11! * 3!) = 12 / 1 = 12

So, the results are:

1. 12! / (9! * 3!) = 1320
2. 12! / (10! * 3!) = 132
3. 12! / (11! * 3!) = 12