1 Answers
Answer: Option 4
ALLAHABAD = 9 letters. Out of these 9 letters there is 4 A's and 2 L's are there. So, permutations = $$\frac{{9!}}{{4!.2!}}$$ = 7560 (a) There are 4 vowels and all are alike i.e. 4A's. _2nd _4th _6th _8th _ These even places can be occupied by 4 vowels. In $$\frac{{4!}}{{4!}}$$ = 1 Way. In other five places 5 other letter can be occupied of which two are alike i.e. 2L's. Number of ways = $$\frac{{5!}}{{2!}}$$ Ways. Hence, total number of ways in which vowels occupy the even places = $$\frac{{5!}}{{2!}}$$ × 1 = 60 ways. (b) Taking both L's together and treating them as one letter we have 8 letters out of which A repeats 4 times and others are distinct. These 8 letters can be arranged in $$\frac{{8!}}{{4!}}$$ = 1680 ways. Also two L can be arranged themselves in 2! ways. So, Total no. of ways in which L are together = 1680 × 2 = 3360 ways. Now, Total arrangement in which L never occur together, = Total arrangement - Total no. of ways in which L occur together. = 7560 - 3360 = 4200 ways