Factorials - Math

! is the symbol for factorial, which means the product of the whole numbers less than the given number.

Thus, .

By definition, .

This is a factorial question. The formula for factorials is .

Simplifying this equation we notice that the 3's, 2's, and 1's cancel so

Alternative Solution

This expression can be simplified because all terms in the expression for 8! are also found in the expression for 10!. By writing the expression below, we are able to cancel 8!.

The cancels out all of except for the parts higher than 4, this leaves a 6 and a 5 left to multilpy

Both the numerator and denominator are factorials. If you expanded both, everything would cancel out except for in the numerator. Multiply those together to get 720.

Thus, since the remaining terms cancel out. 56 is the simplified result.

Remember a number followed by a ! is a factorial. A factorial is the product of the given number and all of the numbers smaller than it down to zero. For example, .

Rather than do all of the math involved for , notice that is the same as

From here, the 's cancel out, leaving us with .

The factorial sign (!) just tells us to multiply that number by every integer that leads up to it. So, can also be written as:

To make this easier for ourselves, we can cancel out the numbers that appear on both the top and bottom:

Simplifying this equation we notice that the 3's, 2's, and 1's cancel so

Alternative Solution

This is a factorial question. The formula for factorials is .

The cancels out all of except for the parts higher than 4, this leaves a 6 and a 5 left to multilpy

The factorial sign (!) just tells us to multiply that number by every integer that leads up to it. So, can also be written as:

To make this easier for ourselves, we can cancel out the numbers that appear on both the top and bottom:

Recall that .

Likewise, .

Thus, the expression can be simplified in two parts:

and

The product of these two expressions is the final answer:

To simplify this, just write out each factorial:

This expression of factorials reduces to (n+1)(n+2). Therefore, the solution must be a number that multiplies to 2 consecutive integers. Only 30 is a product of 2 consecutive integers.

So n would have to be 4 in this problem.

A factorial is a number which is the product of itself and all integers before it. For example

In our case we are asked to divide by . To do this we will set up the following:

We know that can be rewritten as the product of itself and all integers before it or:

Substituting this equivalency in and simplifying the term, we get:

Remember what a factorial is, and first write out what the original equation means. A factorial is a number that you multiply by all whole numbers that come before it until you reach one.

You can simplify because all terms in the expression 17! are found in 20!.

Thus:

Rewrite the factorials in multiplicative order.

In this scenario, the numbers of the factorial in the numerator and denominator CANNOT cancel. Simplify by multiplying out the factorials.