A bouquet contains 16 flowers, each of them either a rose, tulip, daisy, or daffodil. If no two types of flower appear the same number of times in the bouquet, and there are more tulips than any other type of flower, what is the minimum number of tulips?

A. 4
B. 5
C. 6
D. 7
E. 8
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The only possibility that I can see here is tulip should be 6 and either of the three (rose,daisy,daffodil) should be among (5,4,1). This adds upto 6+5+4+1 = 16. Answer should be option C. 6

We need to find the min number of tulips. I interpret this as the least number of tulips with which we can make a 16-flower bouquet. If we have 5 tulips, we can have a maximum of 14 flowers in the bouquet (e.g. 2 roses, 3 daisies, 4 daffodils), so a 16-flower bouquet is not possible. If we have 6 tulips, a 16-flower bouquet is possible (e.g. 1 rose, 4 daisies, 5 daffodils).

Answer: C

Option C

- 16 flowers;
- no two types of flower appear the same number of times in the bouquet;
- more tulips than any other type of flower;

- minimum number of tulips?

x + (x-1) + (x-2) + (x-3) = 16

4x = 22

x = 5,5

5 cannot be the answer (there will be at least 1 other type equal or greater than tulip), so 6 is the answer.

A way to go about it is as follows,
As the number of tulips has to be minimum possible, with the given condition, we can agree that all the flower must be as close to each other as possible (can not be equal) and would have the average value of 4.
(how? 16/4=4)
Now let's divide the 4 numbers into two sets, if a>b>c>d, then a and d are one set and b and c are another
Hence the two numbers (in a set) will have to be at equal distances from 4 in opposite directions to have the same sum as 4+4, ie 4+1, coupled with 4-1 and 4+2 coupled with 4-2.
(Why not 4+0? because 4+0 = 4-0, hence the numbers become equal, which is not allowed)
Therefore clearly the closest (from each other) set of numbers fulfilling this questions conditions are 6,5,3,2
Therefore no. of tulips = 6.
Side note: This method only works because min value is asked, in the case of max, it is better to simply subtract the smallest possible values (1,2,3)
Please do leave a kudos if I could introduce a new method to you