Question

How do you evaluate \[{}^{10}{P_7}\] using a calculator?

Answer 1

Hint: In the given question we have to find the value of \[{}^{10}{P_7}\]using a calculator. Now you should know the formula to expand the given term i.e.
\[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}},\]
Where, \[n! = n(n - 1)(n - 2)......... \times 3 \times 2 \times 1\]

Complete step by step solution:
n the given question we have to find the value of \[{}^{10}{P_7}\] for which we can use the permutation formula i.e.
\[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}},\]
Where, \[n! = n(n - 1)(n - 2)......... \times 3 \times 2 \times 1\]
Now putting the value \[n = 10,r = 7\] we get,
\[
  {}^{10}{P_7} = \dfrac{{10!}}{{\left( {10 - 7} \right)!}} \\
   \Rightarrow {}^{10}{P_7} = \dfrac{{10!}}{{3!}} \\
 \]
Now steps to be followed to solve above term using calculator are:
Step\[1.\] Now in the calculator press \[10\] and then press factorial \[(!)\] sign.
Step\[2.\] Now press the division sign \[( \div )\].
Step \[3.\] Now press \[3\] and then again the factorial \[(!)\] sign.
Step\[4.\] Now press equal to \[( = )\] sign and you will get the result as 604800
Hence, the value of \[{}^{10}{P_7}\] is \[604800\].
So, the correct answer is “\[604800\]”.

Note: Here you should learn the formula used in the above question so that you can solve these types of questions easily. Also learn the value of factorials till \[7!\] so that you can easily calculate the value of larger factorials. Also remember the formula for the combination is little different i.e. \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] .
Here just the \[r!\] is extra in the denominator part so you can derive the permutation formula from this combination formula i.e. \[{}^n{C_r} \times r! = {}^n{P_r}\] this is because permutation is used when you also have to arrange the elements so we just multiply the combination formula by \[r!\].
Also be careful while calculating the factorial don’t miss any term and also while eliminating the terms.