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Question Number 200051 by cortano12 last updated on 13/Nov/23

$$$$$$\mathrm{There}\mathrm{are}\mathrm{many}\mathrm{ways}\mathrm{to}\mathrm{arrange}3\mathrm{red}$$$$\mathrm{balls}\mathrm{and}9\mathrm{black}\mathrm{balls}\mathrm{in}\mathrm{a}\mathrm{circle}$$$$\mathrm{so}\mathrm{that}\mathrm{there}\mathrm{are}\mathrm{a}\mathrm{minimum}\mathrm{of}2$$$$\mathrm{black}\mathrm{balls}\mathrm{between}2\mathrm{adjacent}\mathrm{red}$$$$\mathrm{balls}.$$$$\left(\mathrm{a}\right)180\times 8!\left(\mathrm{b}\right)240\times 7!\left(\mathrm{c}\right)364\times 6!$$$$\left(\mathrm{d}\right)282\times 4!\left(\mathrm{e}\right)144\times 5!$$

Commented by mr W last updated on 13/Nov/23

$$differentoridenticalballs?$$

Commented by cortano12 last updated on 13/Nov/23

$$\mathrm{different}\mathrm{balls}$$

Answered by mr W last updated on 13/Nov/23

$$toarrangetheredballs:2!=2$$$$toarrangetheblackballs:$$$$2+2+5\Rightarrow \frac{3!}{2!}\times 2!\times 2!\times 5!=1440$$$$2+3+4\Rightarrow 3!\times 2!\times 3!\times 4!=1728$$$$3+3+3\Rightarrow 3!\times 3!\times 3!=216$$$$\Rightarrow 2\times (1440+1728+216)=6768✓$$$$\Rightarrow answer\left(d\right)$$

Commented by cortano12 last updated on 13/Nov/23

$$\mathrm{minimum}2\mathrm{red}\mathrm{balls}\mathrm{between}\mathrm{two}$$$$\mathrm{adjacent}\mathrm{black}\mathrm{balls}\mathrm{sir}.$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{possible}3\mathrm{red}\mathrm{balls}\mathrm{between}\mathrm{two}$$$$\mathrm{adjacent}\mathrm{black}\mathrm{balls}?$$

Commented by mr W last updated on 13/Nov/23

$$no,2,3,4or5\mathit{b}\mathit{l}\mathit{a}\mathit{c}\mathit{k}ballsbetween$$$$two\mathit{r}\mathit{e}\mathit{d}ballsasthequestionrequests.$$

Commented by cortano12 last updated on 13/Nov/23

$$\mathrm{l}\mathrm{got}\mathrm{the}\mathrm{answer}\mathrm{A}$$

Commented by cortano12 last updated on 13/Nov/23

Commented by mr W last updated on 13/Nov/23

$$yes,youareright!\left(A\right)iscorrect.$$

Answered by MM42 last updated on 13/Nov/23

$$for‘‘{red}^{″}\to 2$$$$for‘‘{black}^{″}$$$$5,2,2\to 3/4,3,2\to 6/3,3,3\to 1\Rightarrow 10$$$$ans=2\times 10\times 9!=180\times 8!✓$$$$$$

Commented by MM42 last updated on 13/Nov/23

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