Formulaic Maths Problems
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Find the number of combinations of $5$ items from a set of $9$.
Hint: ${}^n \mathrm{C}_r = \dfrac{n!}{r! \, (n-r)!}$
${}^{9} \mathrm{C}_{5} = \dfrac{\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{A1}}\hspace{54px}}~}!}{\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{A2}}\hspace{54px}}~}! \times \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{A3}}\hspace{54px}}~}!}$
${}^{9} \mathrm{C}_{5} = \dfrac{[9]!}{[5]! \times [4]!}$
Cancel and calculate.
Hint: Multiply the numbers down from $n$ in the numerator, and down from $r$ in the denominator.
${}^{9} \mathrm{C}_{5} = \dfrac{\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B1}}\hspace{35px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B2}}\hspace{35px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B3}}\hspace{35px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{B4}}\hspace{35px}}~}}{\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C1}}\hspace{35px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C2}}\hspace{35px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C3}}\hspace{35px}}~} \times \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C4}}\hspace{35px}}~}} = \dfrac{\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{D1}}\hspace{80px}}~}}{\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{D2}}\hspace{80px}}~}} = \class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{DR}}\hspace{80px}}~}$
${}^{9} \mathrm{C}_{5} = \dfrac{[9] \times [8] \times [7] \times [6]}{[4] \times [3] \times [2] \times [1]} = \dfrac{[3024]}{[24]} = [126]$