An anagram of a word is a rearrangement of its
letters. From the word “ALGORITHM”,
a) how many anagrams can be made if each
arrangement begins and ends with a vowel.
b) if four letters are to be selected, how many
these selections contained at least one vowel?
To solve these questions, we need to understand the concept of permutations and combinations.
a) First, let's count the number of vowels in the word "ALGORITHM". We have 4 vowels (A, O, I, and O).
To begin and end with a vowel, we need to fix the positions of the vowels. Since we have 4 vowels, there are 4 possible positions for the first vowel and 1 possible position for the last vowel.
Next, we need to arrange the remaining 7 consonants (L, G, R, T, H, M, and L) between the first and last vowel. The number of ways to arrange these 7 consonants is 7!
Therefore, the total number of anagrams that can be made if each arrangement begins and ends with a vowel is:
4 (choices for the first vowel) * 7! (arrangements of remaining 7 consonants) = 4 * 7! = 20,160.
b) To find the number of selections containing at least one vowel, we need to consider two scenarios:
1. Selecting one vowel and three consonants.
2. Selecting two vowels and two consonants.
1. Selecting one vowel and three consonants:
The number of ways to select one vowel from 4 (A, O, I, and O) is 4C1 = 4.
The number of ways to select three consonants from the remaining 7 consonants is 7C3 = (7 * 6 * 5) / (3 * 2 * 1) = 35.
Using the multiplication principle, the number of selections with one vowel and three consonants is:
4 (choices for one vowel) * 35 (choices for three consonants) = 140.
2. Selecting two vowels and two consonants:
The number of ways to select two vowels from 4 is 4C2 = (4 * 3) / (2 * 1) = 6.
The number of ways to select two consonants from 7 is 7C2 = (7 * 6) / (2 * 1) = 21.
Using the multiplication principle, the number of selections with two vowels and two consonants is:
6 (choices for two vowels) * 21 (choices for two consonants) = 126.
To determine the total number of selections with at least one vowel, we add the results from the two scenarios:
140 (selections with one vowel and three consonants) + 126 (selections with two vowels and two consonants) = 266.
Therefore, the number of selections with at least one vowel is 266.