Question

The digits from 0 to 9 are written on slips of paper and placed in a box. Four of the slips are drawn at random and placed in the order. How many outcomes are possible?

Answer 1

Count the total number of slips formed and consider it as n. Now, consider the number of slips drawn at random as r. Use the formula of combinations ${}^{n}{{C}_{r}}$ to find the numbers of possible ways to draw the required number of slips. Now, considering that the four slips drawn can be in any order, find the number of possible ways to arrange 4 numbers at four places by using the formula $4!$. Finally take the product ${}^{n}{{C}_{r}}\times 4!$ to get the answer.

Complete step by step solution:
Here we have been asked the number of possible outcomes if we have digits from 0 to 9 written on slips of paper, place in a box from which 4 slips are drawn at random and placed in the order in which they are drawn.
Now, total number of slips present = n = 10 (counting from 0 to 9).
Number of slips drawn at random = r = 4.
Therefore number of ways to select 4 slips from a total of 10 slips using the combination formula ${}^{n}{{C}_{r}}$ we have,
$\Rightarrow$ Number of ways of selection = ${}^{10}{{C}_{4}}$
Now, these selected slips are placed in the order in which they are drawn but we don’t know which slips will be selected first so any of the four slips can be selected in any order therefore we can say that these four slips are to be arranged at four places.
$\Rightarrow$ Number of ways to arrange 4 slips at 4 places = $4!$
Since, we have to perform both the functions so we need to take the product to get the total number of outcomes possible.
$\Rightarrow$ Number of outcomes possible = ${}^{10}{{C}_{4}}\times 4!$
On simplifying using the formula ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ we get,
$\therefore$ Number of outcomes possible = 5040

Note: You can solve the question in one step by applying the permutation formula given as ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$. However you must know the difference between the two terms ‘permutation’ and ‘combination’. In general if we have to select r things from n thing then we apply the combinations formula and if after selection we need to arrange those things also then we apply permutation formula.