Need help in following problems related to combinatorial analysis.

Question

1. How many motorcycle number plates can be made if each plate contains 2 different letters followed by 3 different digits?

2. How many four code words are possible using the letters in COMPUTE if (a) the letters may not be repeated? (b) the letters may be repeated?

Answer 1

For the first qquestion: you have to fill in 2 different letters and 3 different digits. In the English alphabets you have 26 letters, so you are choosing 2 from 26 which is $ \binom{26}{2}$. For the numbers you have 10 numbers from 0 to 9 , thus you are choosing 3 from 10, $ \binom{10}{3}$. Hence your answer is $$ \binom{26}{2} \times \binom{10}{3}$$

For the second question, your code consistes of four letters , while COMPUTE contains 7 letters. So for case (a) where letters are not allowed to be repeated, you have to choose 4 from 7, so it is $\binom{7}{4}$. For case (b), where letters in the code are allowed to be repeated, you may use instead of combination arrangement, and write $ A_7^4$.

Hope this helps you. Try with the left 6 questions, and drop a comment for help!

Answer 2

For the first question : There are 26(A-Z) alphabets in total and 10 digits(0-9). Now, let us look into our given case. There are 5 places and we have to fill the first two places with two different alphabets and the next three with 3 different digits.The first place can be filled using any one of the 26 alphabets and the second can be filled using any of the alphabets excluding the one we used to fill the first place i.e 25. Now the third place can be filled using any of the 10 digits, and the fourth can be filled with any of the remaining 9 digits, the fifth place can be filled using any of the remaining digits excluding the two which we used to fill the third and fourth place i.e 8. Thus, First place can be filled in 26 ways, second in 25 ways, third in 10 ways, fourth in 9 ways and fifth in 8 ways. So total number of ways in which the job can be completed = total number of possible number plates = 26x25x10x9x8 =468000 plates.

For the second question : The letter compute consists of 7 digits. The number of four letter words that can be created when repetition is not allowed 7x6x5x4 = 840. (same reason as in first question). Number of four letter words that can created when repetition is allowed = 6x6x6x6 = 1296.