Letters A-J are put into a bag, 5 are selected at random (without replacement) to form a 5letter word:
A. How many different words are possible ?
B. What is the probability the word ENDS with a vowel?
C. What is the probability it starts and ends with a vowel?
A. To find the number of different words that can be formed, we need to calculate the number of ways to arrange the 5 selected letters. Since we are selecting without replacement, the first letter can be any of the 10 letters (A-J). After choosing the first letter, there are only 9 options for the second letter, 8 options for the third letter, 7 options for the fourth letter, and 6 options for the fifth letter. Therefore, the total number of different words possible is:
10 x 9 x 8 x 7 x 6 = 30,240 different words.
B. To calculate the probability that the word ends with a vowel, we need to find the number of words that meet this condition and divide it by the total number of different words possible.
The number of words that end with a vowel can be calculated as follows:
The last letter can be any of the 5 vowels (A, E, I, O, or U). After choosing the last letter as a vowel, there are 9 options for the first letter, 8 options for the second letter, 7 options for the third letter, and 6 options for the fourth letter. Therefore, the total number of words that end with a vowel is:
5 x 9 x 8 x 7 x 6 = 15,120 words
The probability of selecting a word that ends with a vowel is:
15,120 / 30,240 = 0.5 (or 50%)
C. To calculate the probability that the word starts and ends with a vowel, we need to find the number of words that meet this condition and divide it by the total number of different words possible.
The number of words that start and end with a vowel can be calculated as follows:
The first and last letters can each be any of the 5 vowels (A, E, I, O, or U). After choosing the first and last letters as vowels, there are 8 options for the second letter, 7 options for the third letter, and 6 options for the fourth letter. Therefore, the total number of words that start and end with a vowel is:
5 x 8 x 7 x 6 x 5 = 8,400 words
The probability of selecting a word that starts and ends with a vowel is:
8,400 / 30,240 = 0.2778 (or approximately 27.78%)
To answer these questions, you need to consider the number of favorable outcomes and the total number of possible outcomes.
A. How many different words are possible?
To find the number of different words possible, we need to find the total number of different ways you can select 5 letters from the 10 available in the bag. This is calculated using the concept of combinations. The number of ways to select k items from a set of n items is given by the formula nCk, which is the binomial coefficient.
In our case, n (total number of letters) is 10, and k (number of letters selected) is 5. Therefore, the number of different words possible would be 10C5, which can be calculated as:
10C5 = 10! / (5!(10-5)!) = 10! / (5!5!) = (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1) = 252.
Therefore, there are 252 different words possible.
B. What is the probability the word ENDS with a vowel?
To calculate this probability, we need to determine the number of favorable outcomes (words that end with a vowel) and the total number of possible outcomes.
Number of favorable outcomes: We have 10 letters in the bag, and the last letter needs to be a vowel, i.e., A, E, I, or J. So, there are 4 possible vowels for the last position.
Total number of outcomes: We already calculated this in part A, and it is 252.
Therefore, the probability that the word ends with a vowel is 4/252, which can be simplified as 1/63.
C. What is the probability it starts and ends with a vowel?
To calculate this probability, we need to determine the number of favorable outcomes (words that start and end with a vowel) and the total number of possible outcomes.
Number of favorable outcomes: For the first letter, we have 10 letters in the bag, and there are 4 possible vowels. For the last letter, there are also 4 possible vowels. So, the number of favorable outcomes is 4 * 4 = 16.
Total number of outcomes: We already calculated this in part A, and it is 252.
Therefore, the probability that the word starts and ends with a vowel is 16/252, which can be simplified as 2/31.