Find the number of ‘PICTURE’ so that(i) all vowels come together(ii) no two vowels come together скачать в хорошем качестве

Find the number of ‘PICTURE’ so that(i) all vowels come together(ii) no two vowels come together

Find the number of ways of permuting the letters of the word ‘PICTURE’ so that (i) all vowels come together (ii) no two vowels come together In this video, we'll be exploring the different ways of permuting the letters of the word 'PICTURE' while satisfying certain conditions. Specifically, we will be finding the number of ways of permuting the letters such that all vowels come together and such that no two vowels come together. First, let's consider the condition where all vowels come together. The word 'PICTURE' has two vowels, 'I' and 'U', and five consonants, 'P', 'C', 'T', 'R', and 'E'. To satisfy the condition where all vowels come together, we can treat the two vowels as a single letter, say 'V', and then permute the six letters 'V', 'P', 'C', 'T', 'R', and 'E'. We can then arrange the two vowels within the group of six letters in two ways, either 'IV' or 'VI'. Thus, the number of ways of permuting the letters of 'PICTURE' such that all vowels come together is 2 times the number of ways of permuting the six letters, which is 6 factorial. Next, let's consider the condition where no two vowels come together. To satisfy this condition, we need to arrange the two vowels, 'I' and 'U', such that there is at least one consonant between them. We can do this in two ways, either placing 'I' before 'U' or placing 'U' before 'I'. Let's consider the case where 'I' comes before 'U'. We can then think of 'PICTURE' as having the form 'P_C_T_R_E_', where '_' represents a possible location for the vowel 'U'. We need to place the 'I' in one of the five positions, say the first position, and then place the 'U' in one of the four remaining positions such that there is at least one consonant between them. We can think of this as placing the 'U' in one of the four gaps between the consonants or at the end of the string. There are four such gaps, one between 'P' and 'C', two between 'C' and 'T', and one between 'T' and 'R', and there is one end gap after 'E'. Thus, the number of ways of placing the 'U' is 5. Once we have placed the 'U', we can permute the remaining four letters in 4 factorial ways. Thus, the number of ways of permuting the letters of 'PICTURE' such that no two vowels come together is 2 times 5 times 4 factorial. We can obtain the same result by considering the case where 'U' comes before 'I'. The only difference is that we need to place the 'I' in one of the four gaps between the consonants or at the beginning of the string, and there are four gaps and one beginning position available. Thus, the number of ways of permuting the letters of 'PICTURE' such that no two vowels come together is also 2 times 5 times 4 factorial. In this video, we have learned two different ways of permuting the letters of the word 'PICTURE' while satisfying certain conditions. We have found that the number of ways of permuting the letters such that all vowels come together is 2 times 6 factorial, and the number of ways of permuting the letters such that no two vowels come together is 2 times 5 times 4 factorial. These methods can be generalized to other words with different numbers of vowels and consonants and different conditions. JOIN LIVE CLASSES https://chat.whatsapp.com/GmV7W51d8LT... website : www.e-math.in contact : +91 7032507699(whatsapp) visit my website: www.e-math.in youtubechannel:    / onlinemathstutionguntur   facebook page:   / emath2016   twitter :  / e-math2016   whatsapp :+91 7032507699 e-mail : e-math2016@outlook.com #permutation #combinatorics #vowels #consonants #PICTURE #math #mathematics #maths #probability #counting #arrangements #order #factorial #mathvideo #education #learnmath #mathhelp #mathsolutions #mathproblems #mathconcept #youtubemath #youtubemathvideo