Heights Problem

[surajgarg]

8 women of 8 different heights are to pose for a photo in two rows of 4. Each woman in the second row mus stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these instructions are fully adhered to, in how many different ways can the women pose?

A. 2
B. 14
C. 15
D. 16
E. 18

OA later.

[Maciek]

IMO B

We have 2 rows and 4 places in each row

second row EFGH
first row ABCD

The tallest woman and the shortest woman can pose only in one way ( places A and H respectively)

The second tallest woman can take 2 places ( G and D)

The second shortest woman can take 2 places ( B and E)

The third tallest woman can take 3 places ( E, F, and D)

The third shortest woman can take 3 places ( B, C, and E)

The forth tallest woman can take 4 places ( E, F, C and D)

The forth shortest woman can take 4 places ( B, C, D and E)

Let us assign numbers to the women (1,2,3,4,5,6,7,8 - where no. 1 is assigned to the shortest woman and no. 8 to the tallest woman)

Possible solutions:

2468 2478 2568 2578 2678 second row
1357 1356 1347 1346 1345 first row

3468 3478 3568 3578 3678 second row
1257 1256 1247 1246 1245 first row

4568 4578 4678 second row
1237 1236 1235 first row

5678 second row
1234 first row


Thus the women can pose in 14 different ways.

Do you now a faster way to solve this question?

hope it helps!

Best,
Maciek