How many words can be formed combination so that vowels always come together?
Hint: The word daughter has $8$ letters in which $3$ are vowels. For the vowels to always come together consider all the $3$ vowels to be one letter (suppose V) then total letters become $6$ which can be arranged in $6!$ ways and the vowels themselves in $3!$ ways.
Complete step-by-step answer: Show (ii)We have to find the number of words formed when no vowels are together. Note: Combination is used when things are to be arranged but not necessarily in order. Permutation is a little different. In permutation, order is important. Permutation is given by- $ \Rightarrow n! = n(n - 1)(n - 2).......1$ Complete step by step answer: The number of ways the word TRAINER can be arranged so that the vowels always come together are 360. Note: Here while solving such kind of problems if there is any word of $n$ letters and a letter is repeating for $r$ times in it, then it can be arranged in $\dfrac{{n!}}{{r!}}$ number of ways. If there are many letters repeating for a distinct number of times, such as a word of $n$ letters and ${r_1}$ repeated items, ${r_2}$ repeated items,…….${r_k}$ repeated items, then it is arranged in $\dfrac{{n!}}{{{r_1}!{r_2}!......{r_k}!}}$ number of ways. 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 ways vowels come together?(i) Let us suppose 3 vowels as one unit and the number of permutations in E,I,U is3!. Now, we can arrange 4 consonants +1units of vowels in P,I,C,R,EIU =5 ways. So, permutations will be 5!. Hence, the number of permutations on which 3 vowels occur together =5!
Can 2 vowels come together?Since no two vowels can come together, therefore vowels can be inserted in any three places out of the five places available, such as, i.e.,in 5C3 ways, i.e., 10 ways required =24×6×10=1440.
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