Question

Please make detailed answers to the problem. So I can understand fully.

Thank you

Question 6:

A market gardener is planning a planting scheme for the new growing season, and there are 12 different crops to choose from – 8 of which are vegetables and 4 of which are grains.

1. a) Calculate the number of different ways he can select 7 crops to plant, if there are no restrictions.
2. b) Redo (a), with the extra restriction that he must select at most 2 grain crops.
3. c) Say that he has selected the following 7 crops: tomatoes, carrots, peas, radishes, beans, lettuce, cilantro. He wishes to plant these in 7 adjacent rows, with 1 crop in each row. Calculate the number of different ways he can arrange these rows, if there are no other restrictions.
4. d) Redo (c), with the extra restriction that the tomatoes and carrots must be planted in adjacent rows.
5. e) Redo (d), with the further extra restriction that the peas and beans must not be planted in adjacent rows.
6. f) Redo (e), with the further extra restriction that the 7 crops must be planted in a circle, instead of in adjacent rows.

Answer 1

(a)

Selecting 7 crops out of 12 crops is a problem of combination (without permutation, as order of selection is irrelevant).

So, number of ways = = 792

(b)

Out of 7 selected grains, at most 2 can be from grain crops.

So at most 2 grain crops can be selected from 4 grains crops and remaining from 8 vegetable crops.

So, number of ways = = 8*1+28*4+56*6 = 456

(c)

First crop can be planted in any of 7 rows.

Second crop can be planted in any of remaining 6 rows and so on.

So, number of ways = 7! = 5040

(d)

Taking two adjacent rows for tomatoes and carrots as a group we have total 6 items to arrange which can be done in 6! ways.

Further tomatoes and carrots can be planted in two adjacent rows in 2! ways.

So, number of ways = 6! x 2! = 1440

(e)

Number of ways when tomatoes and carrots are planted in adjacent rows = 1440

Number of ways when tomatoes and carrots are planted in adjacent rows as well as peas and beans are planted in adjacent rows = 5! x 2! x 2! = 480

So, number of ways when tomatoes and carrots are planted in adjacent rows but peas and beans are not planted in adjacent rows = 1440 - 480 = 960

(f)

As there is no end in circular arrangement we know, for n objects have (n-1)! circular permutations.

So, number of ways when tomatoes and carrots are planted adjacent = (6-1)!X2! = 240

So, number of ways when tomatoes and carrots are planted adjacent as well as peas and beans are planted

adjacent = (5-1)!x2!x2! = 96

So, number of ways when tomatoes and carrots are planted adjacent but peas and beans are not planted

adjacent = 240-96 = 144