Question

How do you evaluate $ {}^{10}{P_8} $ using a calculator?

Answer 1

Hint: Permutation is an act of arranging the objects or numbers in order. Combinations are the way of selecting the objects or numbers from a group of objects or collections, in such a way that the order of the objects does not matter. This selection of subsets is called a permutation when the order is not a factor. By considering the ratio of the number of desired subsets to the number of all possible subsets. Factorial is defined as the product of all-natural numbers less than or equal to a given natural number.

Complete step-by-step answer:
We will start off by mentioning the definition of $ {}^n{P_r} $ .
We know that $ {}^n{P_r} = \dfrac{{n!}}{{(n - r)!}} $ , where $ n! = n(n - 1)(n - 2)(n - 3)......... \times 1 $ .
Now we will compare our given term with the above-mentioned formula, and evaluate the values of the terms.
So, here
  $
  n = 10 \\
  r = 8 \;
  $
Now apply the formula, and the value of the term $ {}^{10}{P_8} $ .
 \[
  {}^n{P_r} = \dfrac{{n!}}{{(n - r)!}} \\
  {}^{10}{P_8} = \dfrac{{10!}}{{(10 - 8)!}} \\
  {}^{10}{P_8} = \dfrac{{10!}}{{(2)!}} \\
  {}^{10}{P_8} = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \\
  {}^{10}{P_8} = 1814400 \;
 \]
Hence, the value of $ {}^{10}{P_8} $ is \[1814400\] .
So, the correct answer is “ \[1814400\] ”.

Note: While solving such type of questions, make sure that you know your basics well, the terms like factorials and binomial coefficients. You should know what an argument is and what a combination is. Every combinatorial equation or expression will have an interpretation. Also, when you come up with some idea for counting, make sure that you are always counting every possibility.