fair
-sided die is rolled
times and the resulting sequence of
numbers is recorded.
How many different sequences are possible?
How many different sequences consist entirely of even numbers?
How many different sequences are possible if the first, third, and fourth numbers must be the same?
1. There are 6 possible outcomes for each roll, so there are 6^n possible sequences of n rolls.
2. If we want to have all even numbers, we can only roll 2, 4, or 6 on each roll. So there are 3 options for each roll, and the total number of sequences consisting entirely of even numbers is 3^n.
3. If the first, third, and fourth numbers must be the same, we can treat them as one roll and ignore them in the count of possible sequences. So we have 6 options for the first, third, and fourth numbers, and 6 options for the second and fifth numbers, giving us a total of 6^3 * 6^2 = 6^5 possible sequences.
A fair 6-sided die is rolled 4times and the resulting sequence of 4numbers is recorded.
A park bench can seat 3 people. How many seating arrangements are possible if 3 people out of a group of 12 want to sit on the park bench?
To find the answers to these questions, we can use the concept of permutations.
1. How many different sequences are possible?
When rolling a fair-sided die, each roll has 6 possible outcomes (numbers 1 to 6). Since we are rolling the die multiple times, the total number of sequences can be calculated as the product of the number of outcomes for each roll. In this case, since the die is rolled n times, where n is the number of rolls, the total number of sequences is 6^n.
2. How many different sequences consist entirely of even numbers?
For a sequence to consist entirely of even numbers, the die must land on an even number in every roll. Since there are only 3 even numbers (2, 4, and 6) on the die, there are 3 possible outcomes for each roll. Therefore, the number of sequences consisting entirely of even numbers is 3^n.
3. How many different sequences are possible if the first, third, and fourth numbers must be the same?
If the first, third, and fourth numbers must be the same, we have slightly different restrictions for these three dice rolls compared to the remaining rolls. For these three rolls, we only have 1 outcome since they have to be the same. However, for the remaining rolls, we still have 6 outcomes. Therefore, the total number of sequences is 1 * 6 * 6 * 6^(n-3), where n is the number of rolls.
Keep in mind that these calculations assume that the die is fair and all outcomes are equally likely.