Q.1.1 - A First Course in Probability

Question

1. (a) How many different 7-place license plates are possible if the first 2 places are for letters and the other 5 for numbers? (b) Repeat part (a) under the assumption that no letter or number can be repeated in a single license plate 

Step-by-Step Solution

Answer

(a)The ways different 7-place license plates are possible if the first 2 places are for letters and the other 5 for numbers are $26\times 26\times 10\times 10\times 10\times 10\times 10$

(b) Repeat part (a) under the assumption that no letter or number can be repeated in a single license plate  are $26\times 25\times 10\times 9\times 8\times 7\times 6$

Step 1 . Part(a) .Given information

License plates are made by 2 places are for letters and the other 5 for numbers  are given,

We have to find how many different ways 7-place license plates are possible

Step 2. Part(a). Description of explaining ways of the first 2 places are for letters and the other 5 for numbers

By using the generalized basic principle of counting, we can say that there are 26 ways to fill for first place, and also there are 26 ways to fill in second place, so there are $26\times 26$ ways to fill the first two places. Now for a third-place there 10ways and similarly up to 7 ^th place , the number of ways from third places  to 7^th places are $10\times 10\times 10\times 10\times 10$.

Thus the possible ways to fill  the first 2 places are for letters and the other 5 for numbers are $26\times 26\times 10\times 10\times 10\times 10\times 10$

Step 1 . Part(b) .Given information

License plates are made by 2 places are for letters and the other 5 for numbers  are given,

We have to find how many different ways  under the assumption that no letter or number can be repeated in a single license plate  

Step 2. Part(b). Description of explaining ways of no letter or number can be repeated in a single license plate

 If the first place is filled with 26 ways then 2^nd place must be in 25 ways, since no letter or number can be repeated in a single license plate.

For every selection of the first two places, 3^rd  place there are 10 ways to fill in.

For every first three places, ways of fourth place there are 9 ways.

 And the procedure is going on up to the seventh place and there  6 ways to fill in the seventh place,

 thus a number of ways to based on the question are $26\times 25\times 10\times 9\times 8\times 7\times 6$