Factorials are products of positive integers written with an exclamation point, such as \(5!=5\cdot4\cdot3\cdot2\cdot1\). They are commonly used in counting, permutations, combinations, and probability. These problems include evaluating factorials, simplifying factorial fractions, and solving factorial equations by rewriting larger factorials in terms of smaller ones.
Questions
\(\textbf{1)}\) \( 5! \)
The answer is \( 120 \)
\(\,\,\,\,\,\,5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\)
\(\,\,\,\,\,\,5!=120\)
\(\textbf{2)}\) \( 4! \)
The answer is \( 24 \)
\(\,\,\,\,\,\,4!=4 \cdot 3 \cdot 2 \cdot 1\)
\(\,\,\,\,\,\,4!=24\)
\(\textbf{3)}\) \( 1! \)
The answer is \( 1 \)
\(\,\,\,\,\,\,1!=1\)
\(\textbf{4)}\) \( 0! \)
The answer is \( 1 \)
\(\,\,\,\,\,\,0!=1\)
\(\,\,\,\,\,\,\text{By definition, }0!\text{ is equal to }1.\)
\(\textbf{5)}\) \( \frac{16!}{12!4!} \)
The answer is \( 1820 \)
\(\,\,\,\,\,\,\frac{16!}{12!4!}\)
\(\,\,\,\,\,\,=\frac{16\cdot15\cdot14\cdot13\cdot12!}{12!\cdot4!}\)
\(\,\,\,\,\,\,=\frac{16\cdot15\cdot14\cdot13}{4\cdot3\cdot2\cdot1}\)
\(\,\,\,\,\,\,=\frac{43680}{24}\)
\(\,\,\,\,\,\,=1820\)
\(\textbf{6)}\) \( \frac{19!}{18!} \)
The answer is \( 19 \)
\(\,\,\,\,\,\,\frac{19!}{18!}\)
\(\,\,\,\,\,\,=\frac{19\cdot18!}{18!}\)
\(\,\,\,\,\,\,=19\)
\(\textbf{7)}\) \( \frac{8!}{6!} \)
The answer is \( 56 \)
\(\,\,\,\,\,\,\frac{8!}{6!}\)
\(\,\,\,\,\,\,=\frac{8\cdot7\cdot6!}{6!}\)
\(\,\,\,\,\,\,=8\cdot7\)
\(\,\,\,\,\,\,=56\)
\(\textbf{8)}\) \( \frac{n!}{(n-2)!} \)
The answer is \( n(n-1) \)
\(\,\,\,\,\,\,\frac{n!}{(n-2)!}\)
\(\,\,\,\,\,\,=\frac{n(n-1)(n-2)!}{(n-2)!}\)
\(\,\,\,\,\,\,=n(n-1)\)
\(\textbf{9)}\) \( \frac{n!}{n} \)
The answer is \( (n-1)! \)
\(\,\,\,\,\,\,\frac{n!}{n}\)
\(\,\,\,\,\,\,=\frac{n(n-1)!}{n}\)
\(\,\,\,\,\,\,=(n-1)!\)
\(\textbf{10)}\) \( \frac{x}{x!}=50\% \)
The answer is \( x=3 \)
\(\,\,\,\,\,\,\frac{x}{x!}=50\%\)
\(\,\,\,\,\,\,\frac{x}{x!}=\frac{1}{2}\)
\(\,\,\,\,\,\,x!=x(x-1)!\)
\(\,\,\,\,\,\,\frac{x}{x(x-1)!}=\frac{1}{2}\)
\(\,\,\,\,\,\,\frac{1}{(x-1)!}=\frac{1}{2}\)
\(\,\,\,\,\,\,(x-1)!=2\)
\(\,\,\,\,\,\,(x-1)!=2!\)
\(\,\,\,\,\,\,x-1=2\)
\(\,\,\,\,\,\,x=3\)
\(\textbf{11)}\) \( 6! \cdot 7!=x! \)
The answer is \( x=10 \)
\(\,\,\,\,\,\,6! \cdot 7!=x!\)
\(\,\,\,\,\,\,720\cdot5040=x!\)
\(\,\,\,\,\,\,3628800=x!\)
\(\,\,\,\,\,\,10!=3628800\)
\(\,\,\,\,\,\,x=10\)
Challenge Problems
\(\textbf{12)}\)\(16x!+(x+2)!=9(x+1)!\)
The answer is \(x=3\)
\(\,\,\,\,\,\,16x!+(x+2)!=9(x+1)!\)
\(\,\,\,\,\,\,16x!+(x+2)(x+1)x!=9(x+1)x!\)
\(\,\,\,\,\,\,16+(x+2)(x+1)=9(x+1)\)
\(\,\,\,\,\,\,16+x^2+3x+2=9x+9\)
\(\,\,\,\,\,\,x^2+3x+18=9x+9\)
\(\,\,\,\,\,\,x^2-6x+9=0\)
\(\,\,\,\,\,\,\left(x-3\right)^2=0\)
\(\,\,\,\,\,\,x=3\)
\(\textbf{13)}\)\((x+1)!=12(x-1)!\)
The answer is \(x=3\)
\(\,\,\,\,\,\,(x+1)!=12(x-1)!\)
\(\,\,\,\,\,\,(x+1)x(x-1)!=12(x-1)!\)
\(\,\,\,\,\,\,(x+1)x=12\)
\(\,\,\,\,\,\,x^2+x-12=0\)
\(\,\,\,\,\,\,\left(x+4\right)\left(x-3\right)=0\)
\(\,\,\,\,\,\,x=-4\text{ or }x=3\)
\(\,\,\,\,\,\,x=3\text{ since factorial inputs must be nonnegative integers.}\)
\(\textbf{14)}\)\((x+2)!=20x!\)
The answer is \(x=3\)
\(\,\,\,\,\,\,(x+2)!=20x!\)
\(\,\,\,\,\,\,(x+2)(x+1)x!=20x!\)
\(\,\,\,\,\,\,(x+2)(x+1)=20\)
\(\,\,\,\,\,\,x^2+3x+2=20\)
\(\,\,\,\,\,\,x^2+3x-18=0\)
\(\,\,\,\,\,\,\left(x+6\right)\left(x-3\right)=0\)
\(\,\,\,\,\,\,x=-6\text{ or }x=3\)
\(\,\,\,\,\,\,x=3\text{ since factorial inputs must be nonnegative integers.}\)
\(\textbf{15)}\)\(\frac{x!}{(x-2)!}=42\)
The answer is \(x=7\)
\(\,\,\,\,\,\,\frac{x!}{(x-2)!}=42\)
\(\,\,\,\,\,\,\frac{x(x-1)(x-2)!}{(x-2)!}=42\)
\(\,\,\,\,\,\,x(x-1)=42\)
\(\,\,\,\,\,\,x^2-x-42=0\)
\(\,\,\,\,\,\,\left(x-7\right)\left(x+6\right)=0\)
\(\,\,\,\,\,\,x=7\text{ or }x=-6\)
\(\,\,\,\,\,\,x=7\text{ since factorial inputs must be nonnegative integers.}\)
\(\textbf{16)}\)\((x+1)!+x!=144\)
The answer is \(x=4\)
\(\,\,\,\,\,\,(x+1)!+x!=144\)
\(\,\,\,\,\,\,(x+1)x!+x!=144\)
\(\,\,\,\,\,\,x!\left(x+1+1\right)=144\)
\(\,\,\,\,\,\,x!\left(x+2\right)=144\)
\(\,\,\,\,\,\,\text{Test nearby factorials.}\)
\(\,\,\,\,\,\,4!\left(4+2\right)=24\cdot6=144\)
\(\,\,\,\,\,\,x=4\)
\(\textbf{17)}\)\(\frac{(x+2)!}{x!}=30\)
The answer is \(x=4\)
\(\,\,\,\,\,\,\frac{(x+2)!}{x!}=30\)
\(\,\,\,\,\,\,\frac{(x+2)(x+1)x!}{x!}=30\)
\(\,\,\,\,\,\,(x+2)(x+1)=30\)
\(\,\,\,\,\,\,x^2+3x+2=30\)
\(\,\,\,\,\,\,x^2+3x-28=0\)
\(\,\,\,\,\,\,\left(x+7\right)\left(x-4\right)=0\)
\(\,\,\,\,\,\,x=-7\text{ or }x=4\)
\(\,\,\,\,\,\,x=4\text{ since factorial inputs must be nonnegative integers.}\)
\(\textbf{18)}\)\(3(x+1)!=18x!\)
The answer is \(x=5\)
\(\,\,\,\,\,\,3(x+1)!=18x!\)
\(\,\,\,\,\,\,3(x+1)x!=18x!\)
\(\,\,\,\,\,\,3(x+1)=18\)
\(\,\,\,\,\,\,x+1=6\)
\(\,\,\,\,\,\,x=5\)
\(\textbf{19)}\)\(\frac{(x+3)!}{(x+1)!}=56\)
The answer is \(x=5\)
\(\,\,\,\,\,\,\frac{(x+3)!}{(x+1)!}=56\)
\(\,\,\,\,\,\,\frac{(x+3)(x+2)(x+1)!}{(x+1)!}=56\)
\(\,\,\,\,\,\,(x+3)(x+2)=56\)
\(\,\,\,\,\,\,x^2+5x+6=56\)
\(\,\,\,\,\,\,x^2+5x-50=0\)
\(\,\,\,\,\,\,\left(x+10\right)\left(x-5\right)=0\)
\(\,\,\,\,\,\,x=-10\text{ or }x=5\)
\(\,\,\,\,\,\,x=5\text{ since factorial inputs must be nonnegative integers.}\)
\(\textbf{20)}\)\(2x!+(x+1)!=8x!\)
The answer is \(x=5\)
\(\,\,\,\,\,\,2x!+(x+1)!=8x!\)
\(\,\,\,\,\,\,2x!+(x+1)x!=8x!\)
\(\,\,\,\,\,\,x!\left(2+x+1\right)=8x!\)
\(\,\,\,\,\,\,x+3=8\)
\(\,\,\,\,\,\,x=5\)
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