. 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?
take care of the restriction first, placing the vowels...
3XXXXXXX2
the not-used vowel is now part of the remaining 7 letters to go in the places named X
3*7*6*5*4*3*2*1*2 = ....
b) the word "selected" suggests that the order does not matter.
The one we don't want is "all 4 selected are consonants."
Number of selections without restrictions = C(9,4) = 126
number of selections with only consonants = C(6,4) = 15
No number containing at least one vowel = 126-15 = 111
To solve these problems, we can use counting techniques such as permutations and combinations.
a) To find the number of anagrams of the word "ALGORITHM" that begin and end with a vowel, we need to determine the number of ways we can arrange the remaining letters in between the vowels.
1. Identify the vowels in the word: "A", "O", "I"
2. Count the number of vowels: 3
3. Subtract the vowels from the total number of letters in the word: 9 - 3 = 6
4. Rearrange the remaining consonants: 6!
The exclamation mark (!) represents the factorial function, which means multiplying a number by all the positive integers less than itself. In this case, 6! means 6 x 5 x 4 x 3 x 2 x 1 = 720.
5. Multiply the number of rearrangements of the remaining vowels: 3!
Multiplying 3! by 6! will give us the total number of anagrams that begin and end with a vowel:
3! x 6! = 6 x 720 = 4,320
So, there are 4,320 anagrams of the word "ALGORITHM" that begin and end with a vowel.
b) To find the number of selections containing at least one vowel, we need to find the total number of selections and subtract the selections that contain no vowels.
1. Determine the total number of selections: 9C4
In this case, we are selecting 4 letters out of 9 total letters.
The notation "9C4" represents the combination function, which is denoted as "nCr". It calculates the number of ways to choose "r" items from a set of "n" items without considering their order.
Using the combination formula, 9C4 = 9! / (4! * (9 - 4)!) = 126.
2. Determine the number of selections with no vowels: 6C4
In this case, we are selecting 4 letters out of the 6 consonants.
Using the combination formula, 6C4 = 6! / (4! * (6 - 4)!) = 15.
3. Subtract the selections with no vowels from the total selections:
Total selections - Selections with no vowels = 126 - 15 = 111.
So, there are 111 selections containing at least one vowel out of the word "ALGORITHM" when selecting four letters.