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If the word did not have any repeating letters, then this problem would be much simpler. 

For example, let's think of the number of possible arrangements for the not-so-arbitrary word "Melody."

• For the first letter, there are 6 total choices
• For the second letter, there are now 5 possible choices
• For the third letter, there are now 4 possible choices
• This goes on until the final letter where there is only one option. 

Therefore, \(6*5*4*3*2*1\) is the total number of combinations. In shorthand, this can be mathematically expressed with a factorial symbol: \(6!\). 

If there are repetitions in the word, then simply divide by the number of arrangements of the repeated letters. 

Therefore, in "calculator," there are three repeated letters. There are:

• 2 "c's"
• 2 "a's"
• 2 "l's"

Yes, like as what tertre said:

Here's a similar problem, How many ways are there to arrange the word, "Waterfall"

First, count how many letters, and see how many letters in the word repeat, then divide.

Calculator: 10 letters

(a) repeats twice

(c) repeats twice

(l) repeats twice 

So, \(\frac{10!}{2!*2!*2!}=453600\)

Gavin, it appears as if you made a slight error with some basic arithmetic. It happens to all of us.

You correctly identified a special feature of the special 30-60-90 triangle and applied that knowledge correctly: The side length across from the 30° angle is half the length of the hypotenuse. 

You, then, decided to apply your knowledge of the famous Pythagorean Theorem, and this step is where the error is.

I was able to identify this error quickly because of another feature of 30-60-90 triangles. The side length across from the 30° angle multiplied by \(\sqrt{3}\) equals the length of the angle across from the 60º angle. 

A decimal is rational if it terminates or if it repeats indefinitely. The decimal fits the first condition and is, therefore, rational. 

Combinations!

\({5}\choose{1} \)+\({5}\choose{2}\)+\({5}\choose{3}\)= \(5+10+10=25\)

\({6}\choose{2}\)+\({6}\choose{3}\)+\({6}\choose{4}\)=\(15+20+15=50\)

Thus, \(50*25=\boxed{1250}\)

\(\sqrt{1000} = \sqrt{100\cdot10} = \sqrt{100}\cdot\sqrt{10} = 10\sqrt{10}.\)

It's irrational since √10 is irrational since there exists NO integer which, when squared, = 10.

Plotting this on a piece of paper, and labeling all the points;

We have S midway of 8, which is 4

We have T midway of 6, which is 3

Thus, we have a right angle triangle, and we use this formula:

a^2+b^2=c^2

Plugging 4 for a, and 3 for b, we have

4^2+3^2=c^2

c^2=25

\(c=\boxed{5}\)

st=5, (a)

Yes it is rational. 

This is because it can be expressed as a fraction 3.10723 is also 3 10723/100000

note: not all decimals are rational, pi is an example of an irrational decimal.

Here's a similar question: 

https://web2.0calc.com/questions/help_48981

Yes, it is rational; 31.662, and it goes on.

Yes, it is a rational number; decimal.

That is wrong; 3*2+3=9, not 22

This is just combinations, 15C2- \(\frac{15!}{2!*13!}=\frac{15*14}{2*1}=15*7=\boxed{105}\)

Gavin is correct! You have to put the brackets around 1/3, not just 1. So, \((\frac{1}{3})^4=\frac{1}{81}\)

That looked harder so let me tell you my way: 

3*22=66 

easy peasy lemon squeasy 

thanks. it did help.

Two quintets 

\((\frac{1}{3})^4=\frac{1^4}{3^4}=\frac1{81}\) 

I hope this helped,

Gavin

This is a simple choosing problem. 

15 choose 2 can be expressed as:

\(\binom{15}{2}=\frac{15\cdot14}{2\cdot1}=105\)

i hope this helped,

Gavin

Once again, we use the 30 - 60 -90 rule. 

The hypotenuse is twice the length of the side opposite of the 30 degree angle.

Therefore the side opposite of the 30 degree angle is 12/2 = 6 

We can use the Pythagorean theorem, to find the length of a. 

12^2 - 6^2 = a^2

a^2 = 108

a = √108 = 6√3

I hope this helped,

Gavin

In a 30 - 60 - 90, triangle, the side opposite of the 30 degree angle is half of the hypotenuse.

We don’t have to treat this problem as 3D geometry, we can just pay attention to the side facing us. Now it is just a 2D geometry problem. 

Using the rule I stated at the top of this post, the hypotenuse is twice the side opposite of the 30 degree angle, which is 3 meters long. 

Therefore, the hypotenuse is 3*2 = 6 

Since we know 2 sides of a right triangle, we can find the third. 

using the Pythagorean theorem, x^2 = 6^2 - 3^2 = 30 

x = √30

I hope this helped,

Gavin

1:   A right square pyramid with base edges of length 8sqrt2 units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane?