7 cars are parked side-by-side. 4 of them are black. How many different ways are there of arranging the cars so that the 4 black cars are parked next to each other?
- print Print
- list Cite
Expert Answers
calendarEducator since 2012
write511 answers
starTop subjects are Math, Science, and Business
We have 7 cars parked side-by-side and 4 of them are black. To work out the number of ways the black cars can be parked next to each other, we proceed as follows:
First, we need to use a formula for choosing r items in a row out of n (r<=n).
This is n - r + 1
Because there is the first way, eg in the case of 4
b b b b . . .
and then n - r more ways as we shift to the right by 1 place each time
So, we have n = 7 spaces and r = 4 black cars. The number of ways of getting 4 black cars in a row is
7 - 4 +1 = 4 ways
Secondly, we need to think of the number of ways the individual 4 black cars can be arranged amongst themselves given that 4 black cars are arranged in 4 particular spaces.
This is the number of permutations of the 4 black cars. This is equal to
4! = 4 x 3 x 2 x 1 = 24
There are 4 ways of placing the first car and 3 ways of placing the second, 2 ways of placing the third and then only 1 way of placing the fourth.
For each possible arrangement of 4 black cars in a row there are 24 ways of arranging the individual cars amongst themselves, so in total there are
24 * (n - r + 1) = 24 * (7 - 4 + 1) = 24 * 4 = 96 ways
Answer: there are 96 ways
Related Questions
- Calculate the mean and standard deviatio of he amount of revenue each car generates.When parking...
- 1 Educator Answer
- How many sides and angles are on a square, circle, triangle, and rectangle?
- 1 Educator Answer
- Two cars are driving towards each other from towns that are 150 miles apart. The first car is...
- 1 Educator Answer
- The annual income of A and B are in the ratio 3: 4 and their annual expenditure are in the ratio...
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.