 Hi and how are you all today? The question says how many words with or without meaning can be formed using all the letters of the word equation in a time so that the waffle or consonants occur together. Now here we will be using the formula for number of permutations and that is NPR is equal to N factorial divided by N minus R factorial and the usage of this formula will be the key idea of our question. Let us proceed on with the solution. Now here the given word is equation. Now to find the number of words so that vowels and consonants occur together here in this word equation has five vowels that is E, U, A, I and O whereas there are three different consonants that is Q, T and N. Let us take all vowels in a single unit and all consonants in another single unit. Now using term one of permutation there will be as many words found in all vowels as one letter and all consonants as one letter as permutation of two different letters taken two at a time without repetition so the required number of permutations is equal to and let it be the first equation. Now the unit containing the five vowels will have five different arrangement itself thus there will be as many words found in all five vowels are their permutation of five different letters at five different times without repetition so required number of permutation when the five vowels will have taken in five different time that is equal to five p five and let it be the second equation. Similarly the unit containing the consonants there are three consonants and there will be three different arrangements for it so here also required number of permutation is equal to three p three let this be the third equation and applying the multiplication principle the number of words where all vowels and consonants together equal to first equation multiplied by the second equation multiplied by the third equation that we have made above that is equal to two p two multiplied by five p five multiplied by three p three on using the formula we have two factorial divided by two minus two factorial multiplied by five factorial five minus five factorial multiplied by three factorial divided by three minus three factorial further we have two factorial multiplied by five factorial multiplied by three factorial which is two multiplied by one five multiplied by four three two one multiplied by three factorial that is equal to one multiplied by 120 multiplied by six which gives us the answer as one four four zero so this is our required answer take care bye for now