How many arrangements of the word ALGORITHM begin with a vowel and end with a consonant?
there are 3 vowels and 6 consonants.
So there are 3 ways to fill the first spot and 6 ways to fill the last.
The leaves the remaining 7 letters to go anywhere else.
so
3x7x6x5x4x3x2x1x6
= 3(7!)(6) = 90720
there are 3 vowels and 6 consonant
arrangements starting with vowel A
=(7*6*5*4*3*2*1)*6
=30240
arrangements starting with vowel O
= 302460
arrangements starting with vowel I
=30240
: . total arrangements =30240+302460+30240
=90720 arrangements
reiny can u answer my other two questions
System.out.println("bc" + 2 + 3)
Well, if we want to arrange the word "ALGORITHM" so that it begins with a vowel and ends with a consonant, we first need to identify which letters are vowels and which are consonants.
In this case, the vowels are A, O, and I, while the consonants are L, G, R, T, H, and M.
Since our condition is that it must begin with a vowel and end with a consonant, we have 3 options for the first letter and 6 options for the last letter.
The remaining 8 letters can be arranged in 8! (8 factorial) ways.
So, the total number of arrangements would be:
3 (choices for the first vowel) * 8! (arrangements of the remaining letters) * 6 (choices for the last consonant) = 145,152
Remember, this is assuming that repeating letters are indistinguishable. So, if we consider repeating letters as distinguishable, the number of arrangements would be different. Either way, I hope this helps lighten up the mathematical mood a little!
To determine the number of arrangements of the word "ALGORITHM" that begin with a vowel and end with a consonant, we first need to count the total number of arrangements of the word.
The word "ALGORITHM" consists of 9 letters, including the vowels 'A', 'O', and 'I', and the consonants 'L', 'G', 'R', 'T', 'H', and 'M'.
To find the total number of arrangements, we can use the formula for permutations of a set with repeated elements.
In this case, the letter 'L' occurs twice, and the letters 'A', 'O', and 'I' each occur once. The remaining letters are all distinct.
The formula for permutations with repeated elements is given by:
Total number of arrangements = n! / (r1! * r2! * ... * rk!)
Where n is the total number of objects, and r1, r2, ..., rk are the repetitions of each respective element.
In our case, n = 9, r1 = 2 (for 'L'), r2 = 1 (for 'A'), r3 = 1 (for 'O'), r4 = 1 (for 'I'), and the rest of the letters are distinct (k = 0).
Substituting these values into the formula:
Total number of arrangements = 9! / (2! * 1! * 1! * 1!)
Calculating further:
Total number of arrangements = 362,880 / (2 * 1 * 1 * 1)
Hence, the total number of arrangements of the word "ALGORITHM" is 181,440.
Now, to determine the number of arrangements that begin with a vowel ('A', 'O', or 'I') and end with a consonant ('L', 'G', 'R', 'T', 'H', or 'M'), we need to consider the positions of the first and last letters.
Since we have three vowels and six consonants, the number of arrangements that start with a vowel is 3 * 8!, which accounts for the three different vowels at the first position and the remaining eight letters being rearranged.
Similarly, the number of arrangements that end with a consonant is 6 * 8!, accounting for the six different consonants at the last position and the remaining eight letters being rearranged.
To find the number of arrangements that satisfy both conditions, we multiply the two results:
Number of arrangements = 3 * 6 * 8!
Calculating further:
Number of arrangements = 3 * 6 * 40,320
Hence, the number of arrangements of the word "ALGORITHM" that begin with a vowel and end with a consonant is 725,760.