No.	Problems & Solutions [Click Hide/Show to display the solutions below the question]
02.	In a random arrangement of the letters of the word “UNCOPYRIGHTABLE”. Find (a) the probability and odds that all the vowels come together and (b) the probability that the vowels do not come together

Solution

Number of Letters/Characters in the word “UNCOPYRIGHTABLE” = 15 {U, N, C, O, P, Y, R, I, G, H, T, A, B, L, E} In the experiment of testing for the number of words that can be formed using the letters of the word “UNCOPYRIGHTABLE” Total No. of Possible Choices = Number of words that can be formed using the 15 letters of the word “UNCOPYRIGHTABLE” ⇒ n = 15P15 (Or) 15! Let "A" be the event of the word formed with all the vowels coming together "B" be the event of the word formed with all the vowels not coming together For Event "A" No. of vowels (a) = 5 {U, O, I, A, E} No. of other letters (Consonants) = 10 {N, C, P, Y, R, G, H, T, B, L} Considering all the vowels as a unit No. of letters = 11 {N, C, (U, O, I, A, E), P, Y, R, G, H, T, B, L} No. of places to be filled to form a word = 11 Number of Favourable/Favorable Choices = The no. of words that can be formed using the letters of the word “UNCOPYRIGHTABLE” such that all the vowels are together. ⇒ mA = (No. of ways in which the 11 places can be filled with the 11 letters       [considering the 5 vowels as a unit]) × (No. of ways in which the vowels can be interarranged between       themselves) = 11P11 × 5P5 = 11! × 5! Probability of forming the word using all the letters of the word “UNCOPYRIGHTABLE” such that all the vowels are together ⇒ Probability of occurrence of Event "A" = Number of Favourable/Favorable Choices for the Event Total Number of Possible Choices for the Experiment ⇒ P(A) = mA n = 11! × 5! 15! = 11! × 5 × 4 × 3 × 2 × 1 15 × 14 × 13 × 12 × 11! = 1 3 × 7 × 13 = 1 273 • Odds No. of UnFavorable Choices = Total No. of possible choices − No. of Favorable choices ⇒ mAc = n − mA = 15! − 11! × 5! = (15 × 14 × 13 × 12 × 11!) − (11! × 5 × 4 × 3 × 2 × 1) = 11! (32,760 − 120) = 11! (32,640) » in favor/favour Odds in Favour of the words formed with the letters of the word “UNCOPYRIGHTABLE” having all the vowels together ⇒ Odds in Favor/Favour of Event "A" = No. of Favourable Choices : No. of UnFavorable Choices = mA : mAc = 11! × 5! : 11! (32,640) = 120 : 32,640 = 1 : 272 » against Odds against the words formed with the letters of the word “UNCOPYRIGHTABLE” having all the vowels together ⇒ Odds against Event "A" = No. of UnFavourable Choices : No. of Favorable Choices = mAc : mA = 11! (32,640) : 11! × 5! = 32,640 : 120 = 272 : 1 For Event "B" Number of Favourable/Favorable Choices = The no. of words that can be formed using the letters of the word “UNCOPYRIGHTABLE” such that all the vowels are not together. ⇒ mB = (Total number of words that can be formed with the 15 letters       of the word “UNCOPYRIGHTABLE” ) − (Number of words that can be formed with all the vowels together) = mAc = 11! (32,640) Probability of forming the word using all the letters of the word “UNCOPYRIGHTABLE” such that all the vowels are not together ⇒ Probability of occurrence of Event "B" = Number of Favourable/Favorable Choices for the Event Total Number of Possible Choices for the Experiment ⇒ P(B) = mB n = 11! × (32,640) 15! = 11! × 32,640 15 × 14 × 13 × 12 × 11! = 272 273 For Event "B" [Alternative] Probability of forming the word using all the letters of the word “UNCOPYRIGHTABLE” such that all the vowels are not together ⇒ Probability of occurrence of Event "B" = Probability of non-occurrence of Event "A" ⇒ P(B) = P(Ac) = 1 − P(A) = 1 − 1 273 = 273 − 1 273 = 272 273

Author Credit : The Edifier