I. In how many ways can the first, second and third positions be taken by candidates in a test, assuming that there must be no tie.
How many candidates are there?
To determine the number of ways the first, second, and third positions can be taken by candidates without any tie, you can use the concept of permutations.
In a permutation, the order of objects matters. Since there must be no tie, each candidate must occupy a distinct position.
To calculate permutations, you can use the formula for the number of permutations of n objects taken r at a time:
P(n, r) = n! / (n - r)!
where n! represents n factorial, which means the product of all positive integers from 1 to n.
In this case, we need to determine the number of ways three candidates can be chosen for the positions of the first, second, and third places. Since there are three positions in total, we want to find P(3, 3).
Using the formula:
P(3, 3) = 3! / (3 - 3)!
= 3! / 0!
= 3!
The factorial of 3 is calculated as 3! = 3 x 2 x 1 = 6.
Therefore, the number of ways the first, second, and third positions can be taken, assuming no tie, is 6.