Determine the number \(7!\) represents.

Determine the number \(4(5!)\) represents.

Determine the number \(\displaystyle{\frac{5}{6!}}\) represents.

\(\displaystyle{\frac{5}{6!} = \frac{1}{144}}\)

Determine the number \(\displaystyle{\frac{5}{6!}}\) represents.

\(\begin{array}{rcl} \displaystyle{\frac{5}{6!}} & = & \displaystyle{\frac{5}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}} \\ & = & \displaystyle{\frac{1}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 6}} \\ & = & \displaystyle{\frac{1}{144}} \end{array} \)

Determine the number \(\displaystyle{6!-\frac{(7-3)!}{3!}}\) represents.

Simplify the factorial \(\displaystyle{\frac{8!}{6!}}\).

Simplify the factorial \(\displaystyle{\frac{10!}{7!3!}}\).

Simplify the factorial \(\displaystyle{\frac{11!}{8!}}\).

Simplify the factorial \(\displaystyle{\frac{10!}{8!2!}}\).

Simplify the factorial \(\displaystyle{\frac{20!}{18!}}\).

\(\displaystyle{\frac{20!}{18!} = 380}\)

Simplify the factorial \(\displaystyle{\frac{20!}{18!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{20!}{18!}} & = & \displaystyle{\frac{18! \cdot 19 \cdot 20}{18!} } \\ & = & 19 \cdot 20 = 380 \end{array} \)

Simplify the factorial \(\displaystyle{\frac{13!}{16!}}\).

\(\displaystyle{\frac{13!}{16!} = \frac{1}{3360}}\)

Simplify the factorial \(\displaystyle{\frac{13!}{16!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{13!}{16!}} & = & \displaystyle{\frac{13!}{13! \cdot 14 \cdot 15 \cdot 16}} \\ & = & \displaystyle{\frac{1}{14 \cdot 15 \cdot 16} = \frac{1}{3360}} \end{array} \)

Simplify the factorial \(\displaystyle{\frac{23!}{5!20!}}\).

\(\displaystyle{\frac{23!}{5!20!} = \frac{1771}{20}}\)

Simplify the factorial \(\displaystyle{\frac{23!}{5!20!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{!23}{5!20!}} & = & \displaystyle{\frac{20! \cdot 21 \cdot 22 \cdot 23}{20! \cdot 2 \cdot 3 \cdot 4 \cdot 5}} \\ & = & \displaystyle{\frac{7 \cdot 11 \cdot 23}{4 \cdot 5}} \\ & = & \displaystyle{\frac{1771}{20}} \end{array} \)

Simplify the factorial \(\displaystyle{\frac{(7+2)!}{6!2!}}\).

\(\displaystyle{\frac{(7+2)!}{6!2!} = 252}\)

Simplify the factorial \(\displaystyle{\frac{(7+2)!}{6!2!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{(7+2)!}{6!2!}} & = & \displaystyle{\frac{9!}{6!2!}} \\ & = & \displaystyle{\frac{6! \cdot 7 \cdot 8 \cdot 9}{6! \cdot 2}} \\ & = & \displaystyle{\frac{7 \cdot 8 \cdot 9}{2}} \\ & = & \displaystyle{\frac{7 \cdot 4 \cdot 9}{1} = 252} \end{array} \)

Simplify the factorial \(\displaystyle{\frac{5!(4-3)!}{4!}}\).

\(\displaystyle{\frac{5!(4-3)!}{4!} = 5}\)

Simplify the factorial \(\displaystyle{\frac{5!(4-3)!}{4!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{5!(4-3)!}{4!}} & = & \displaystyle{\frac{4! \cdot 5 \cdot 1!}{4!}} \\ & = & 5 \cdot 1 = 5 \end{array} \)

Simplify \(\displaystyle{ \frac{3! 7!}{6! 4!} }\)

\( 7/4 \)

Simplify \(\displaystyle{ \frac{3! 7!}{6! 4!} }\)

Simplify \(\displaystyle{ \frac{12! 7!}{9! 6!} }\)

\( 12 \cdot 11 \cdot 10 \cdot 7 = 9240 \)

Simplify \(\displaystyle{ \frac{12! 7!}{9! 6!} }\)

Simplify \(\displaystyle{ \frac{10! 6!}{8! 7!} }\)

\( 90/7 \)

Simplify \(\displaystyle{ \frac{10! 6!}{8! 7!} }\)

Simplify the factorial \(\displaystyle{\frac{(n-3)!}{n!}}\).

\(\displaystyle{\frac{(n-3)!}{n!} = \frac{1}{(n-2)(n-1)n}}\)

Simplify the factorial \(\displaystyle{\frac{(n-3)!}{n!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{(n-3)!}{n!}} & = & \displaystyle{\frac{(n-3)!}{(n-3)! \cdot (n-2) \cdot (n-1) \cdot n}} \\ & = & \displaystyle{\frac{1}{(n-2) \cdot (n-1) \cdot n}} \end{array} \)

Simplify the factorial \(\displaystyle{\frac{(2n+1)!}{(2n)!}}\).

\(\displaystyle{\frac{(2n+1)!}{(2n)!} = (2n+1)}\)

Simplify the factorial \(\displaystyle{\frac{(2n+1)!}{(2n)!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{(2n+1)!}{(2n)!}} & = & \displaystyle{\frac{1 \cdot 2 \cdot . . . \cdot (2n-1) \cdot (2n) \cdot (2n+1)}{1 \cdot 2 \cdot . . . \cdot (2n-1) \cdot (2n)}} \\ & = & \displaystyle{\frac{(2n)! \cdot (2n+1)}{(2n)!} = (2n+1)} \end{array} \)

Simplify the factorial \(\displaystyle{\frac{(3n-1)!}{(3n+1)!}}\).

\(\displaystyle{\frac{(3n-1)!}{(3n+1)!} = \frac{1}{(3n)(3n+1)}}\)

Simplify the factorial \(\displaystyle{\frac{(3n-1)!}{(3n+1)!}}\).

\(\begin{array}{rcl} \displaystyle{\frac{(3n-1)!}{(3n+1)!}} & = & \displaystyle{\frac{(3n-1)!}{(3n-1)! \cdot (3n) \cdot (3n+1)}} \\ & = & \displaystyle{\frac{1}{(3n)(3n+1)}} \end{array} \)

Simplify the factorial \(\displaystyle{\frac{k!}{(k-2)!}}\).

Simplify \(\displaystyle{ \frac{(n+1)!}{(n-2)!} }\)

\( (n+1)n(n-1) = n^3 - n \)

Simplify \(\displaystyle{ \frac{(n+1)!}{(n-2)!} }\)

Although this instructor multiplies out the factors, check with your instructor to see if they require you to do that.

Simplify \(\displaystyle{ \frac{(n+2)!}{(n-1)!} }\)

\( (n+2)(n+1)n = n^3 + 3n^2 + 2 \)

Simplify \(\displaystyle{ \frac{(n+2)!}{(n-1)!} }\)

Check with your instructor to see if they want you to multiply out your answer.

Simplify \(\displaystyle{ \frac{(n+1)!}{(n+3)!} }\)

\(\displaystyle{ \frac{1}{(n+3)(n+2)} }\)

Simplify \(\displaystyle{ \frac{(n+1)!}{(n+3)!} }\)

Check with your instructor to see if they want you to multiply out the denominator.

Simplify \(\displaystyle{ \frac{n!}{(n-2)!} }\)

\( n(n-1) \)

Simplify \(\displaystyle{ \frac{n!}{(n-2)!} }\)

Check with your instructor to see if they want you to multiply out your answer.

Simplify \(\displaystyle{ \frac{(2m)!}{(2m+2)!} }\)

\(\displaystyle{ \frac{1}{(2m+2)(2m+1)} }\)

Simplify \(\displaystyle{ \frac{(2m)!}{(2m+2)!} }\)

Check with your instructor to see if they want you to multiply out the denominator.

Simplify \(\displaystyle{ \frac{(n+3)!}{n!} }\)

\( (n+3)(n+2)(n+1) \)

Simplify \(\displaystyle{ \frac{(n+3)!}{n!} }\)

Although he says at the end of the problem that you need to foil it out, most instructors will not require you to do that for these types of problems.

Simplify \(\displaystyle{ \frac{(3n+2)!}{(3n-1)!} }\)

\( (3n+2)(3n+1)(3n) \)

Simplify \(\displaystyle{ \frac{(3n+2)!}{(3n-1)!} }\)

Again, do not multiply out the answer unless specifically told to do so by your instructor.

Simplify \(\displaystyle{ \frac{(n^2-4)!}{(n-2)(n^2-5)!} }\)

The last factorial in the denominator applies only to the \((n^2-5)\) term. So the denominator can be written more clearly as \((n-2)[(n^2-5)!]\).

Simplify \(\displaystyle{ \frac{(n^2-4)!}{(n-2)(n^2-5)!} }\)

\(\displaystyle{ \frac{n^2-4}{n-2} }\)

Simplify \(\displaystyle{ \frac{(n^2-4)!}{(n-2)(n^2-5)!} }\)

The last factorial in the denominator applies only to the \((n^2-5)\) term. So the denominator can be written more clearly as \((n-2)[(n^2-5)!]\).

Although they give the final answer as \(n+2\), you know from the domain and range page that they should have also said that \( n \neq 2 \) as part of the final answer.

Simplify \(\displaystyle{ \frac{(n!)^2}{(n-1)!(n+1)!} }\)

\(\displaystyle{ \frac{n}{n+1} }\)

Simplify \(\displaystyle{ \frac{(n!)^2}{(n-1)!(n+1)!} }\)

Simplify \(\displaystyle{ \frac{(n+3)!n!}{(n+2)!(n-1)!} }\)

\(n(n+3)\)

Simplify \(\displaystyle{ \frac{(n+3)!n!}{(n+2)!(n-1)!} }\)

Check with your instructor to see if they want you to multiply out your answer.

Simplify \(\displaystyle{ \frac{(n+2)!-n!}{(n+1)!} }\)

\( n+2 - 1/(n+1) \)

Simplify \(\displaystyle{ \frac{(n+2)!-n!}{(n+1)!} }\)

Simplify the factorial \(\displaystyle{\frac{(k+2)!}{(k-1)!}}\).

This problem is solved in two separate videos by two different instructors. Watching both of them will help you better understand these concepts.