a. How many ways can five letters of the word TRIANGLE be arranged? b. HOw many

Question

a. How many ways can five letters of the word TRIANGLE be arranged?
b. HOw many ways can the letters of the word TRIANGLE be arranged if the first three letters must be RAN(in any order), and the last letter must be a vowel?
c. How many different ways can the letters of the word TRIANGLE be arranged if the vowels IAE cannot be changed, though their placement may (IAETRNGL and TRIANGEL are acceptable but EIATRNGL and TRIENGLA are not).
d. How many different sets of five letters can be formed from the word TRIANGLE if we want at least one vowel?

Explanation / Answer

Note that all eight letters are different. a) Five letters can be arranged in P(8,5) = 8*7*6*5*4 = 6720 b) The first 3 letters must be RAN in any order, so there are 3! possibilities. There are two other vowels, so there are 2 possibilities for the last letter. Then, there are 4! choices for letters 4-7 equals 6*2*4! = 12*24=288. c) I would guess that the 5 other letters may be in any order, so this is 5!. Since order matters for the 3 vowels, there are C(8,3) orders they may be put in. Thus, 8*7*6/(5*4*3)*5! = 8*7*6*2=672 d) The quickest way to do this is take all 5 letter combinations, and subtract the ones with 5 consonants. P(8,5)-P(5,5) = 8*7*6*5*4-5! = 6600

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