 Hi and how are you all today? I am Priyanka. The question says how many words with or without meaning each of two vowels and three consonants can be formed from the letter of the word daughter. Let us start with our solution. Now the word given to us is equal to daughter. Right? Now here we have three vowels that is a, u and e whereas we have consonants that is d, g, h, t. Right? Now here we need to find out the number of combinations that can be done with all objects taken at the time. Now the formula that will be helping us in doing so that is for finding out the number of combination that is ncr is equal to n factorial divided by r factorial multiplied by n minus r factorial. Now this formula is the key idea of our question. Let us proceed on. Now here there will be as many ways of selecting two vowels in the question. We are required to make it by two vowels and three consonants having meaning or without meaning. So the required number of selections is equal to three out of two. So it will be three c two. Let us take it as the first equation. Proceeding on. There will be as many ways of selecting three consonants as there are combination of five consonants taken three at a time. So required number of ways of selection of three consonants is equal to out of five we are selecting three let it be the second equation. Now on applying the multiplication principle or the formula that has been mentioned in it. Here we will be applying the multiplication principle the number of combinations of two vowels and three consonants is equal to equation one multiplied by equation two that is three c two multiplied by five c three. Now two vowels and three consonants these five letters also can be arranged among themselves in a number of ways. Permutation of five different letters taken five at a time without repetition. So these five can also be taken together. So here we have number of permutations that is five p five and let this be the third equation. Now on applying the multiplication principle we have number of words formed by using two vowels three consonants is equal to equation one multiplied by equation two multiplied by equation three that is c two multiplied by five c three multiplied by this can be written as five factorial right. Now on applying the formula of combinations we have n factorial that is three so we have three factorial divided by two factorial multiplied by three minus two factorial. Similarly five factorial divided by three factorial five minus three factorial multiplied by five factorial again. Now on solving it we have three factorial divided by two factorial one factorial multiplied by five factorial divided by three factorial multiplied by two factorial multiplied by five factorial three factorial will get cancelled out with each other. Let us further have we have five factorial can be written as five multiplied by four multiplied by three multiplied by two factorial and since we are having two five factorials so it can be multiplied by two and here we have two two factorials or it can be further simplified here only that is multiplied by four multiplied by three multiplied by two multiplied by one for one five factorial multiplied by five four three two one and in the denominator we have two multiplied by one multiplied by one factorial can be written as one only two factorial is two multiplied by one further on cancelling out we are left with five multiplied by four multiplied by three again five multiplied by four multiplied by three that is five multiplied by four gives us 20 20 multiplied by three it gives us 60 multiplied by 60 gives us 3600 so this is our required answer take care