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distinct object distinct group group size fixed group sizes are equal

by | Dec 10, 2021 | Uncategorized | 0 comments

case (1) → (i) → (B)Distinct object Distinct group(i) Group size fixed(B) Group sizes are equal
422DistinctGroup sizes are equalDistinct

number of ways of division =  \frac{{4!}}{{2!2!}} =  \frac{{4 \times 3}}{2} = 6

BUT WAIT Let do it manually

Let's objects are a, b, c, dDivide into two-two groupsab cdac bdad bcbc adbd accd abSo actual number of Division is 3i.e. = 3But when Group sizes are distinct431a, b, c, da bcdb cdac dabd abc4= = 4

Let’s take another example:-

a, b, c, d, e, fa, b, c, d, e, f624633 = 20ab cdefac bdefad bcefae bcdfaf bcdebc adefbd acefbe acdfbf acdecd abefce abdfcf abdede abcfdf abceef abcdTotal = 15abc defabd cefabe cdfabf cdeacd beface bdfacf bdeade bcfadf bceaef bcdGroups more than these are repeating groupsHence total number of ways of division = 10i.e. actual number of waysof division = × = 10
Example-1022222Objects are distinctGroups are distinct but group sizes are sameDivision × Distribution

Example:- Distribute 52 cards among 4 players equally.
Solution:-

5213131313Objects are DistinctDivisonDistributionDistinct players (Groups)but group sizes are same

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