15. Here is a surprising use of sequences to answer a counting question: Howmany license plates consist of 6 symbols, using only the three numerals 1, 2, and3 and the four letters a, b, c, and d, so that no numeral appears after any letter?For example, "31ddac", "123321", and "ababab" are each acceptable licenseplates, but "13ba2c" is not.a. First answer this question by considering different cases: how many ofthe license plates contain no numerals? How many contain one numeral,etc.b. Now use the techniques of this section to show why the answer is 47 – 37.

Answered: 15. Here is a surprising use of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The given condition has no numeral appear after any letter can be understood as if license plate consist of numerals , then the numerals occupy the starting position and only after using the desired number of numerals , letters can be used .

Let us consider the problem in following cases :

Case 1: No numeral used

If License plate consist of no numerals ,

Then all the 6 symbols will be letters .

There are 4 letters .

So, each of 6 position in the symbol will be occupied by these 4 letters.

Since letters can be repeated , there are 4 ways to fill each of these 6 positions .

That is, $4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^{6}$ ways to fill the license plate in this case .

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