Finding number of arrangements on a shelf with one book of a kind in middle

Question

There are 6 Novels and 3 Dictionaries. There are to be made arrangements of books in a Shelf consisting of 4 Novels and 1 Dictionary, such that Dictionary is always in the middle.

I tried to do this by: There should be 2 novels to the left of dictionary and 2 to the right i.e. 6C2 and out of remaining 4C2, and dictionaries can be arranged as 3C1.

So total arrangements: 6C2 * 4C2 * 3C1 = 15*6*3 = 270? Is my reasoning correct? Can you tell me where I am wrong, Since the answer is incorrect. Thanks in advance

Are the novels distinct? Does the order of the novels matter? — Keatinge, Jul 03 '18 at 14:44
@N.F.Taussig I see, I will keep that in mind. So, I just have to use the permutations formula? 6P4 * 3? — MR. Raindrop, Jul 03 '18 at 15:41

Key Flex · Accepted Answer · 2018-07-03T15:31:42.080

2

Select $1$ dictionary from $3$ in $^3C_1$ ways

Select $4$ novels from $6$ novels in $^6C_4$ ways

Since you chose $4$ novels from $6$, now you can arrange those $4$ novels in $4!$ ways.

Since the dictionary should be in middle $NNDNN$ is the arrangement

$$^6C_4\times^3C_1\times4!\times1!=1080\mbox{ ways.}$$

edited Jul 03 '18 at 15:31

answered Jul 03 '18 at 15:22

Key Flex

9,475
7
17
34

is the 4! the arrangement of novels among themselves? – MR. Raindrop Jul 03 '18 at 15:29
Yes, $4!$ is the arrangement of novels among themselves. See the edit – Key Flex Jul 03 '18 at 15:32

Finding number of arrangements on a shelf with one book of a kind in middle

1 Answers1