How many words can be formed from TRIANGLE so vowels always come together?
Total number of words in ‘TRIANGLE’ = 8 Show Out of 5 are consonants and 3 are vowels If vowels are not together, taken we have the following arrangement V | C | V | C | V | C | V | C | V | C | V Consonant can be arranged in 5! = 120 ways Vowel occupy 6 places ∴ 3 vowels can be arranged in 6 places = 6P3 = `(6!)/((6 - 3)!)` = `(6!)/(3!)` = 120 ways So, the total arrangement = 120 × 120 = 14400 ways Here, the required arrangement = 14400 ways. First of all, $\text{TRIANGLE}$ has $8$ distinct letters, $3$ of which are vowels($\text{I, A, E}$) and rest are consonants($\text{T, R, N, G, L}$). While attempting this, I came up with the idea of putting alternate vowels and consonants not to group same types together. So, I decided to form two 'batteries'. [$\text{V}$ stands for Vowels and $\text{C}$ stands for consonants.] $$\text{V} \text{ C}\text{ V} \text{ C}\text{ V} $$ And, $$\text{C} \text{ V}\text{ C} \text{ V}\text{ C}$$ If we count all the permutations and then add them up (Mutually Exclusive Events), we can get total number of permutations. Now, For the first case, $3$ vowels can be arranged in the $3$ spaces required in $3! = 6$ ways From $5$ consonants, $2$ spaces can be filled with consonants in $^5P_2 = 20$ ways One battery, $(8 - 3- 2) = 3$ letters to arrange. Total number of permutations : $6 * 20 * 4! = 2880$. In Second case, From $3$ vowels, $2$ spaces can be filled with vowels in $^3P_2 = 6$ ways From $5$ consonants, $3$ spaces can be filled with consonants in $^5P_3 = 60$ ways. One battery, $(8 - 2- 3) = 3$ letters to arrange. Total number of permutations : $6 * 60 * 4! = 8640$ So, Total number of permutations for the word $\text{TRIANGLE} = 2880 + 8640 = 11520$ Again, My answer is incorrect, according to my textbook. They report $14400$ is the correct answer. So, what did I miss here now? Please elaborate, and I'll be happy with any sort of help. [Seriously, this morning is getting even more hectic for me] Hint: In this question, we first need to find the total number of arrangements possible with the given letters of the word using the permutation formula given by \[{}^{n}{{P}_{r}}\]. Then we need to find the number of words in which two vowels are together but first selecting the two vowels and then arranging all the letters using the formula \[{}^{n}{{C}_{r}}\]. Now, find the number of words in which 3 vowels are together and then subtract 2 vowels together from total words and add 3 vowels together.Complete step by step solution: Note: How many words can be formed from the letters of the word TRIANGLE so that vowels always come together?Solution : For`'`TRIANGLE`'`there are 8 letters
if T and E fixed in starting than total possible ways will be`6!` `6! =6*5*4*3*2*1=720` ways. How many words can be formed with the word TRIANGLE?Therefore, the total number of words formed from the word TRIANGLE is 40,320 out of which 720 words start with T and end with E.
How many ways combine can be arranged so that vowels always together?The number of ways the word TRAINER can be arranged so that the vowels always come together are 360.
How many words can be formed combination vowels are together?The vowels (EAI) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720.
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