suppose that 3 digits are selected at random from the set S={1,2,3,4,5,6} and are arranged in random order. Find the probability that the resulting 3-digit number is less than 300.
To find the probability that the resulting 3-digit number is less than 300, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
Since we are selecting 3 digits from the set S={1, 2, 3, 4, 5, 6} and arranging them in random order, the total number of possible outcomes is given by the number of permutations of 3 items taken from a set of 6, which is denoted as P(6, 3) and calculated as:
P(6, 3) = 6! / (6 - 3)!
= 6! / 3!
= 6 * 5 * 4
= 120
So, there are 120 possible outcomes.
Number of favorable outcomes:
Since we want the resulting 3-digit number to be less than 300, we need to consider the following cases:
Case 1: The first digit is 1.
In this case, the remaining two digits can be any two from the set {2, 3, 4, 5, 6}. So, the number of favorable outcomes is P(5, 2) calculated as:
P(5, 2) = 5! / (5 - 2)!
= 5! / 3!
= 5 * 4
= 20
Case 2: The first digit is 2.
In this case, the second digit can be any one from the set {0, 1}. The third digit can be any one from the set {1, 2, 3, 4, 5, 6}. So, the number of favorable outcomes is equal to the number of elements in the set {0, 1} multiplied by the number of elements in the set {1, 2, 3, 4, 5, 6}, which is given by 2 * 6 = 12.
So, the total number of favorable outcomes is 20 + 12 = 32.
Probability calculation:
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability = Number of favorable outcomes / Total number of possible outcomes
= 32 / 120
= 8 / 30
= 4 / 15
Therefore, the probability that the resulting 3-digit number is less than 300 is 4/15.
To find the probability that the resulting 3-digit number is less than 300, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of outcomes:
Since we are selecting 3 digits at random from the set S={1,2,3,4,5,6}, there are 6 choices for the first digit, 6 choices for the second digit, and 6 choices for the third digit. Hence, the total number of outcomes is 6 * 6 * 6 = 216.
Number of favorable outcomes:
To have a 3-digit number less than 300, the first digit must be either 1 or 2.
If the first digit is 1, then there are 5 choices for the second digit (2, 3, 4, 5, 6) and 6 choices for the third digit (any of the 6 digits remaining in the set S).
If the first digit is 2, then there is only 1 choice for the second digit (1) and 6 choices for the third digit.
Hence, the number of favorable outcomes is (5 * 6) + (1 * 6) = 30 + 6 = 36.
Therefore, the probability that the resulting 3-digit number is less than 300 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
Probability = Number of favorable outcomes / Total number of outcomes
Probability = 36 / 216
Probability = 1/6
So, the probability is 1/6.
Can digits be repeated?
I will assume they CANNOT
number of possible 3 digit numbers
= 6*5*4 = 120
to be less than 300, the number must start with either a 1 or a 2
number of such cases = 2*5*4 = 40
prob of event = 40/120 = 1/3