SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the method of getting possible options.
When asked to find the number of possible options, this is a permutation problem. Therefore, we use the permutation formula.
[tex]^nP_r=\frac{n!}{(n-r)!}[/tex]STEP 2: Solve the first question
[tex]\begin{gathered} ^nP_r=\frac{n!}{(n-r)!} \\ For\text{ the first question, n=}45,r=3 \\ ^{45}P_3=\frac{45!}{(45-3)!}=\frac{45!}{42!}=\frac{45\times44\times43\times42!}{42!} \\ \Rightarrow45\times44\times43=85140 \end{gathered}[/tex]There are 85140 possible options.
STEP 3: Solve the second question
[tex]\begin{gathered} ^nP_r=\frac{n!}{(n-r)!} \\ n=7,r=5 \\ ^7P_5=\frac{7!}{(7-5)!}=\frac{7!}{2!}=2520 \end{gathered}[/tex]There are 2520 possible options.
STEP 4: Solve the third question.
[tex]\begin{gathered} ^nP_r=\frac{n!}{(n-r)!} \\ n=11,r=9 \\ ^{11}P_9=\frac{11!}{(11-9)!}=\frac{11!}{2!}=19958400 \end{gathered}[/tex]There are 19958400 possible options.
STEP 5: Solve the fourth question
[tex]\begin{gathered} ^nP_r=\frac{n!}{(n-r)!} \\ n=20,r=4 \\ ^{20}P_4=\frac{20!}{(20-4)!}=\frac{20!}{16!}=\frac{20\times19\times18\times17\times16!}{16!}=20\times19\times18\times17=116280 \end{gathered}[/tex]There are 116280 possible options
