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Question Number 124826 by bramlexs22 last updated on 06/Dec/20

How many ways are there to arrange the letters of the word ′ VISITING′ if no two I′s are adjacent ?

$$\:{How}\:{many}\:{ways}\:{are}\:{there}\:{to}\:{arrange}\: \\ $$$${the}\:{letters}\:{of}\:{the}\:{word}\:'\:{VISITING}' \\ $$$${if}\:{no}\:{two}\:{I}'{s}\:{are}\:{adjacent}\:? \\ $$

Answered by mr W last updated on 06/Dec/20

Method I arrange at first the five other letters and then the three I′s in 6 positions: □V□S□T□N□G□ 5!×C_3 ^6 =2400 Method II arrange at first the three I′s and then the five other letters: □I■I■I□ □ =zero or more other letters ■ =one or more other letters (1+x+x^2 +..._(□) )^2 (x+x^2 +x^3 +..._(■) )^2 =(x^2 /((1−x)^4 )) =x^2 Σ_(k=0) ^∞ C_3 ^(k+3) x^k coef. of term x^5 : k=3 ⇒C_3 ^6 ⇒C_3 ^6 ×5!=2400

$${Method}\:{I} \\ $$$${arrange}\:{at}\:{first}\:{the}\:{five}\:{other}\:{letters} \\ $$$${and}\:{then}\:{the}\:{three}\:{I}'{s}\:{in}\:\mathrm{6}\:{positions}: \\ $$$$\Box\mathrm{V}\Box\mathrm{S}\Box\mathrm{T}\Box\mathrm{N}\Box\mathrm{G}\Box \\ $$$$\mathrm{5}!×{C}_{\mathrm{3}} ^{\mathrm{6}} =\mathrm{2400} \\ $$$$ \\ $$$${Method}\:{II} \\ $$$${arrange}\:{at}\:{first}\:{the}\:{three}\:{I}'{s}\:{and} \\ $$$${then}\:{the}\:{five}\:{other}\:{letters}: \\ $$$$\Box\mathrm{I}\blacksquare\mathrm{I}\blacksquare\mathrm{I}\Box \\ $$$$\Box\:={zero}\:{or}\:{more}\:{other}\:{letters} \\ $$$$\blacksquare\:={one}\:{or}\:{more}\:{other}\:{letters} \\ $$$$\left(\underset{\Box} {\mathrm{1}+{x}+{x}^{\mathrm{2}} +...}\right)^{\mathrm{2}} \left(\underset{\blacksquare} {{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} +...}\right)^{\mathrm{2}} \\ $$$$=\frac{{x}^{\mathrm{2}} }{\left(\mathrm{1}−{x}\right)^{\mathrm{4}} } \\ $$$$={x}^{\mathrm{2}} \underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}{C}_{\mathrm{3}} ^{{k}+\mathrm{3}} {x}^{{k}} \\ $$$${coef}.\:{of}\:{term}\:{x}^{\mathrm{5}} : \\ $$$${k}=\mathrm{3}\:\Rightarrow{C}_{\mathrm{3}} ^{\mathrm{6}} \\ $$$$\Rightarrow{C}_{\mathrm{3}} ^{\mathrm{6}} ×\mathrm{5}!=\mathrm{2400} \\ $$

Commented by john_santu last updated on 06/Dec/20

great method II

$${great}\:{method}\:{II} \\ $$

Commented by malwan last updated on 06/Dec/20

I want to master method II like you mrW thank you

$${I}\:{want}\:{to}\:{master}\:{method}\:{II} \\ $$$${like}\:{you}\:{mrW} \\ $$$${thank}\:{you} \\ $$

Commented by mr W last updated on 06/Dec/20

it′s a very powerful method, try to apply it!

$${it}'{s}\:{a}\:{very}\:{powerful}\:{method},\:{try}\:{to} \\ $$$${apply}\:{it}! \\ $$

Answered by john_santu last updated on 06/Dec/20

Commented by bramlexs22 last updated on 06/Dec/20

great!

$${great}! \\ $$

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