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Ways in which you can arrange the 4 different flowers = 4!
Ways in which you can arrage the set of 4 flowers +the rest of the flowers = 5!

Although I always get confused with this questions. If it doesn't say that the 4 flowers must be in the same order, can I assume that I can flip them around? Don't know why I always get confused by this.

The number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated is: A) 4!4! B) 288 C) 8!/4! D) 5!4! E) 8!4!

4 flowers which are always together can be considered as one SET,

Therefore we have to arrange one SET ( 4 flowers ) and 4 other flowers into a garland.

Which means, 5 things to be arranged in a garland.

(5-1)!

And the SET of flowers can arrange themselves within each other in 4! ways.

Therefore

(5-1)!*(4!)

But, Garland, looked from front or behind does not matter. Therefore the clockwise and anti clockwise observation does not make difference.

That means clock-wise and anti-clock wise combinations differ from each other. Hmmmm.....perhaps I should visualize more! I just imagined the flowers look the same, whether looked from front of behind in a garland. But their colors would differ!

1. We have 5 different things: the group of 4 flowers and 4 separate flowers. 5P5=5! 2. to arrange the group of 4 flowers we have 4P4=4! ways. So, 5!*4! 3. circular symmetry means that variants with "circular shift" are the same variant. We can make 5 "circular shifts". Therefore, N=5!*4!/5=4!*4!
_________________

The number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated is: A) 4!4! B) 288 C) 8!/4! D) 5!4! E) 8!4!

[1234]5678

Assume that 1234 are alwasy together So. we can arrange themselves in 4! ways. X5678 Now treat [1234]=X one single group we have 5 flower snad arrange in circular way= (5-1)!

4!*4! A.
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Your attitude determines your altitude Smiling wins more friends than frowning

The number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated is: A) 4!4! B) 288 C) 8!/4! D) 5!4! E) 8!4!

I feel 4! * 4! is not the final answer to this question. This number should be divided by 2 because a single garland when turned around gives us a different arrangement, but its still the same garland.

if 4 flowers must be toghter, we can think that at first we must seat that flowers in 5 seats, in that case ther are 5! cases, but we have 4flowers which in every case of 5! we can arrange its in 4! case, so there are 5!*4! cases Answer is D

I feel 4! * 4! is not the final answer to this question. This number should be divided by 2 because a single garland when turned around gives us a different arrangement, but its still the same garland.

Answer: 4! * 4!/2 = 288 B

This is a good point.

There are two cases of circular-permutations:

1. If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by \((n-1)!\).

2. If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by \(\frac{(n-1)!}{2!}\).

Specific garland (as I understand) when turned around has different arrangement, but its still the same garland as Samrus pointed out. So clock-wise and anti-clock-wise orders are taken as not different.

Hence we'll have the case 2: \(\frac{(5-1)!*4!}{2}=288\)
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