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DETAILED SOLUTION Now, we have been given 8 chairs which are numbered from 1 to 8. Also, it has been given that women choose the chairs from amongst the chairs marked 1 to 4, and then men select from remaining chairs. In total there are 2 women and three men who wish to occupy one chair each. Now, we know the number of ways of selecting r objects among n is <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript n subscript r" width="27" height="19" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«none/»«mi»n«/mi»«/mmultiscripts»«/math»'/>. So, we have the ways in which we can choose two chairs among four numbered 1 to 4 is <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript 4 subscript 2" width="28" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«none/»«mn»4«/mn»«/mmultiscripts»«/math»'/> and we can arrange the women then in 2! ways. Also, we have the ways of selecting 3 chairs among the rest 6 chairs is <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript 6 subscript 3" width="28" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«none/»«mn»6«/mn»«/mmultiscripts»«/math»'/> and in them we can permute the men in 3! ways. So, in total we have number of possible arrangements as, <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript 4 subscript 2" width="28" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»2«/mn»«none/»«mprescripts/»«none/»«mn»4«/mn»«/mmultiscripts»«/math»'/> 2! <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAKCAYAAABSfLWiAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAAC1JREFUeNpjYMAE1VDMQKIcUYpJMgCbJrIMQDZoNyUGUMUQir1DccBSLYrpDwAMJw7rHduE6QAAAE50RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4RDc7PC9tbz48L21hdGg+HjBxXgAAAABJRU5ErkJggg==" class="Wirisformula" alt="cross times" width="17" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#215;«/mo»«/math»'/> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAVCAYAAABVAo5cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAARlJREFUeNpjYEAAcyA+CMQ/gPg/FNMMaAHxOSDWYaATWAjEtgx0BK+BOBqIHwLxLSAOpbWFf4B4NhDzATEblJ2IR70SEHcB8QUg/gZ1aDMQCxJrIchXTEh8Hqgh2EAJEG+BRgFMjywQFwDxfGItnI3mOpAvz2BRVw5VSzHQgyYcUJCyAPEUIPZAU6MGxHeh8lQB0VADX+JIND3QYKMbuAbEKvSyjAlaAtENgBLRJwY6gy9AzEWkWlhZ/AuaV93IsXAuNA+SCjTw5Gm8QB6I3wFxJVKe5QBiH6hjXPDE/ztyg1UFml8/QYPsAxCvAWIvHOqFoXk6kR5xDovHWHomNFDJ1QrEafRO4XTNw7Y4KgKaxB/IZzugKRwMAMGVOYuIdMBSAAAAmHRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbXVsdGlzY3JpcHRzPjxtaT5DPC9taT48bW4+MzwvbW4+PG5vbmUvPjxtcHJlc2NyaXB0cy8+PG5vbmUvPjxtbj42PC9tbj48L21tdWx0aXNjcmlwdHM+PC9tYXRoPlpcTrcAAAAASUVORK5CYII=" class="Wirisformula" alt="C presuperscript 6 subscript 3" width="28" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mn»3«/mn»«none/»«mprescripts/»«none/»«mn»6«/mn»«/mmultiscripts»«/math»'/> 3! Now, we know that  <img src="data:image/png;base64,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" class="Wirisformula" alt="C presuperscript n subscript r" width="27" height="19" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»C«/mi»«mi»r«/mi»«none/»«mprescripts/»«none/»«mi»n«/mi»«/mmultiscripts»«/math»'/> = <img src="data:image/png;base64,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" class="Wirisformula" alt="P presuperscript n subscript r space. space r factorial" width="50" height="19" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»P«/mi»«mi»r«/mi»«none/»«mprescripts/»«none/»«mi»n«/mi»«/mmultiscripts»«mo»§#160;«/mo»«mo».«/mo»«mo»§#160;«/mo»«mi»r«/mi»«mo»!«/mo»«/math»'/> Therefore, we have, Total ways = <img src="data:image/png;base64,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" class="Wirisformula" alt="P presuperscript 4 subscript 2" width="26" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»P«/mi»«mn»2«/mn»«none/»«mprescripts/»«none/»«mn»4«/mn»«/mmultiscripts»«/math»'/> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAKCAYAAABSfLWiAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAJufgqOgAAAC1JREFUeNpjYMAE1VDMQKIcUYpJMgCbJrIMQDZoNyUGUMUQir1DccBSLYrpDwAMJw7rHduE6QAAAE50RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+JiN4RDc7PC9tbz48L21hdGg+HjBxXgAAAABJRU5ErkJggg==" class="Wirisformula" alt="cross times" width="17" height="10" style="vertical-align:-1px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#215;«/mo»«/math»'/> <img src="data:image/png;base64,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" class="Wirisformula" alt="P presuperscript 6 subscript 2" width="26" height="21" style="vertical-align:-5px" role="math" data-mathml='«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mmultiscripts»«mi»P«/mi»«mn»2«/mn»«none/»«mprescripts/»«none/»«mn»6«/mn»«/mmultiscripts»«/math»'/>

Eight chairs are numbered from 1 to 8. Two women and three men wish to occupy one chair each. First women choose the chairs from amongst the chairs marked 1 to 4; and then the men select the chairs from the remaining. The number of possible arrangements is-

Solution for the question - eight chairs are numbered from 1 to 8. two women and three men wish tooccupy one chair each. first women choose the chairs from amon