By Rudolf Kingslake

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**Additional info for Applied optics and optical engineering,Vol.II**

**Example text**

If repeated letters are allowed ? Solution. Here n is 4 and r is 3, so the number of such words is 43 or 64. Problem 47. How many “words” of three distinct letters can be formed from the letters of the word MAST ? Solution. The number is 4P3 = 4! or (4 − 3) ! 4! or 24. 1! Problem 48. How many distinguishable permutations of the letters in the word BANANA are there ? Solution. We begin by tagging the A’s and N’s in order to distinguish between them temporarily. For the letters B, A1, N1, A2, N2, A3, there are 6 !

24. Once these five men have sat on alternate chairs, the five women can occupy the 5 empty chairs in 5 ! ways. Thus, total number of ways, in this case, will be 24 * 5 ! = 24 * 120 = 2880. 60. Find the sum of all the four-digit numbers that can be formed with the digits 3, 2, 3 and 4. Solution. One thing is worth noticing here that a four-digit number so formed does not conatin a repeated digit except the digit 3. This is implied from the question, because if it were not so, 3 should not have been repeated in the list of the digits.

E) First, we observe that relative positions in the box are immaterial so that order does not count. Therefore, this is a combination problem Secondly, a box might consist of a dozen of one variety of doughnut, so that we see that this problem allows unlimited repetitions. The answer then is C(8 – 1 + 12, 12) = C(19, 12). 107. In how many ways can the letters of the word attention be rearranged ? COUNTING PRINCIPLES AND GENERATING FUNCTIONS 41 Solution. The word attention has nine letters, three of one kind, two of another, and four other different letters.

### Applied optics and optical engineering,Vol.II by Rudolf Kingslake

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