Difference between revisions of "2017 AMC 8 Problems/Problem 5"

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==Solution 3==
 
==Solution 3==
  
First, we evaluate 1+2+3+4+5+6+7+8, to get 36. We notice that 36 is 6 squared, so we can go like <math>\frac{\cdot 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{6 \cdot 6}</math> then cancel the 6s out, to get <math>\frac{\cdot 4 \cdot 5 \cdot 7 \cdot 8}{1}</math>. Now that we have escaped fraction form, multiplying \cdot 4 \cdot 5 \cdot 7 \cdot 8. Multiplying these, we get <math>\boxed{\textbf{(B)}\ 1120}</math>.
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First, we evaluate 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8, to get 36. We notice that 36 is 6 squared, so we can go like <math>\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{6 \cdot 6}</math> then cancel the 6s out, to get <math>\frac{4 \cdot 5 \cdot 7 \cdot 8}{1}</math>. Now that we have escaped fraction form, multiplying <math>4 \cdot 5 \cdot 7 \cdot 8 \cdot</math>. Multiplying these, we get <math>\boxed{\textbf{(B)}\ 1120}</math>.
  
 
==Video Solution==
 
==Video Solution==

Revision as of 01:20, 6 November 2020

Problem 5

What is the value of the expression $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{1+2+3+4+5+6+7+8}$?

$\textbf{(A) }1020\qquad\textbf{(B) }1120\qquad\textbf{(C) }1220\qquad\textbf{(D) }2240\qquad\textbf{(E) }3360$

Solution 1

Directly calculating:

We evaluate both the top and bottom: $\frac{40320}{36}$. This simplifies to $\boxed{\textbf{(B)}\ 1120}$.

Solution 2

It is well known that the sum of all numbers from $1$ to $n$ is $\frac{n(n+1)}{2}$. Therefore, the denominator is equal to $\frac{8 \cdot 9}{2} = 4 \cdot 9 = 2 \cdot 3 \cdot 6$. Now we can cancel the factors of $2$, $3$, and $6$ from both the numerator and denominator, only leaving $8 \cdot 7 \cdot 5 \cdot 4 \cdot 1$. This evaluates to $\boxed{\textbf{(B)}\ 1120}$.

Solution 3

First, we evaluate 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8, to get 36. We notice that 36 is 6 squared, so we can go like $\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8}{6 \cdot 6}$ then cancel the 6s out, to get $\frac{4 \cdot 5 \cdot 7 \cdot 8}{1}$. Now that we have escaped fraction form, multiplying $4 \cdot 5 \cdot 7 \cdot 8 \cdot$. Multiplying these, we get $\boxed{\textbf{(B)}\ 1120}$.

Video Solution

https://youtu.be/cY4NYSAD0vQ

~icematrix2

See Also

2017 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AJHSME/AMC 8 Problems and Solutions

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