There space 6 English books, 4 scientific research books, 7 magazines, and 3 invernessgangshow.netematics books. In how plenty of ways can you arrange the shelf if:a) English and also Science books are indistinct?b) English books should be together?

Pls someone assist me ~ above this one. Thank you!

EDIT: i actually have actually an early stage answer. Because that a, $(10!)/(6! 4!)$ methods for the English and also Science books, climate multiply to $10!$ (ways because that the others)? Is this correct? I"m actually not certain if I recognize the border in a correctly.

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The number of different permutations of $n$ objects, whereby $n_1$ space of one kind, $n_2$ room a different kind $\dots$ and also there are $k$ various kinds is:$$\fracn!n_1! \times n_2! \times \dots \times n_k!$$

In how numerous ways have the right to $6$ English books, $4$ scientific research books, $7$ magazines, and also $3$ invernessgangshow.netematics publications be arranged on a shelf if English publications are indistinguishable and also science books are indistinguishable?

We have actually a full of $6 + 4 + 7 + 3 = 20$ books. Select six that the $20$ positions for the English books and four the the continuing to be $14$ positions because that the scientific research books. The staying ten positions deserve to be filled with books and magazines in $10!$ ways.

$$\binom206\binom14410! = \frac20!6!14! \cdot \binom14!4!10! \cdot 10! = \frac20!6!4!$$

In her attempt, you did not take right into account the total variety of positions ~ above the shelf.

In how many ways can $6$ English books, $4$ scientific research books, $7$ magazines, and $3$ invernessgangshow.netematics books be arranged on a shelf if English publications should it is in together?

If all the publications are intended come be unique (switching the stimulate of the concerns would have made this clearer), treat the English books as a solitary object, therefore we have actually $1 + 4 + 7 + 3 = 15$ objects to arrange. Then multiply by the number of ways of arranging the 6 English books within the block of English books.

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If we room still claimed to law the English publications as being indistinguishable and the science publications as being indistinguishable, choose six the the $15$ positions for the science books, one of the staying $8$ positions for the block of English books, then arrange the magazines and also invernessgangshow.netematics books in the staying positions.

I believe the very first of these 2 interpretations is intended, yet I would have reversed the stimulate of the questions to make that clear.