In how many ways can the letters of the word GRADIENT be arranged so that two...

 

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In how many ways can the letters of the word GRADIENT be arranged so that two vowels are not placed together?" To solve this, we can use a combinatorial approach where we first arrange the consonants and then place the vowels in the gaps between consonants ensuring that no two vowels are adjacent. The word "GRADIENT" has 8 letters with 3 vowels (A, I, E) and 5 consonants (G, R, D, N, T). Here are the steps to calculate the number of arrangements: 1. Arrange the 5 consonants in a row, which can be done in 5! ways. 2. Find the number of spaces between the consonants where the vowels can go. There will be 6 spaces (before the first consonant, between each pair of consonants, and after the last consonant). 3. Place each vowel in a separate space. This can be done in P(6, 3) ways, which is the permutation of 6 items taken 3 at a time. The total number of arrangements will be the product of the number of ways to arrange the consonants and the number of ways to arrange the vowels in separate spaces. Let's calculate this. In how many ways can the letters of the word GRADIENT be arranged so that two vowels are not placed together?" To solve this, we can use a combinatorial approach where we first arrange the consonants and then place the vowels in the gaps between consonants ensuring that no two vowels are adjacent. The word "GRADIENT" has 8 letters with 3 vowels (A, I, E) and 5 consonants (G, R, D, N, T). Here are the steps to calculate the number of arrangements: 1. Arrange the 5 consonants in a row, which can be done in 5! ways. 2. Find the number of spaces between the consonants where the vowels can go. There will be 6 spaces (before the first consonant, between each pair of consonants, and after the last consonant). 3. Place each vowel in a separate space. This can be done in P(6, 3) ways, which is the permutation of 6 items taken 3 at a time. The total number of arrangements will be the product of the number of ways to arrange the consonants and the number of ways to arrange the vowels in separate spaces. Let's calculate this.