 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that What is the probability that in the arrangement of the alphabet of the word permutation the vowels will always occur together? Let us start with the solution of the given question. In this question we have to find the probability that in the arrangement of the alphabets of the word permutation the vowels will always occur together. We know that probability of an event is given by favourable outcomes upon total number of outcomes. First we shall find total number of outcomes. Here total number of outcomes will be the total number of arrangements of the alphabets of the word permutation. Here we see that there are 11 alphabets and letter t is coming twice. We know that number of arrangements of n things in which n1 objects are of one kind, n2 objects are of other kind and so on such that n1 plus n2 plus and so on up to nk is equal to n Then the number of distinguishable permutations of these objects is given by n factorial upon n1 factorial into n2 factorial into and so on up to nk factorial. So here in the word permutation we have n is equal to 11 and here t appears twice so n1 will be equal to 2. So number of ways in which alphabets of the word permutation can be arranged will be given by 11 factorial upon 2 factorial. Now favourable outcomes will be number of arrangements of alphabets when vowels are together. In this word there are 5 vowels that is a, e, i, o and u. Now let us consider these vowels as one alphabet. So now total number of alphabets will be 1, 2, 3, 4, 5, 6 and 7. Now here we are treating these vowels as one alphabet and all these vowels can be permuted among themselves in 5 factorial ways. Now by fundamental principle of counting if there are 2 events a and b where event a can occur in m ways and event b can occur in n ways then these events can occur in m into n ways. So here number of arrangements will be equal to 7 factorial into 5 factorial because these 7 alphabets can be arranged in 7 factorial ways and these vowels among themselves can be arranged in 5 factorial ways so total number of arrangements will be 7 factorial into 5 factorial. But we can see that here t is repeated twice so number of arrangements will be 7 factorial into 5 factorial upon 2 factorial so number of favourable outcomes will be 7 factorial into 5 factorial upon 2 factorial and total number of outcomes are given by 11 factorial upon 2 factorial so probability that vowels occur together favourable outcomes that is 7 factorial into 5 factorial by 2 factorial whole upon total number of outcomes that is 11 factorial upon 2 factorial so this is equal to 7 factorial into 5 factorial upon 2 factorial into 2 factorial upon 11 factorial. Now here 2 factorial cancels with 2 factorial and here we are left with 7 factorial into 5 factorial upon 11 factorial. Now this is equal to 7 factorial into now 5 factorial can be written as 5 into 4 into 3 into 2 into 1 whole upon 11 factorial can be written as 11 into 10 into 9 into 8 into 7 factorial and now 7 factorial cancels with 7 factorial and this is equal to now here 2 into 1 is 2 and 2 into 4 is 8 4 cancels with 4 3 into 1 is 3 and 3 into 3 is 9 5 into 1 is 5 and 5 into 2 is 10 so this is equal to 1 upon 11 into 2 into 3 that is 66 thus the probability that vowels occur together is given by 1 upon 66. This is the required answer. This completes our session. Hope you enjoyed this session.