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# show that the number of ways if selecting n−object out of 3n−objects, n of which are like and rest different is 22n−1+(2n)!2(n!)2

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## show that the number of ways if selecting n−n−object out of 3n−3n−objects, nn of which are like and rest differnt is 22n−1+(2n)!2(n!)2 solution:Accordingtoques:wehave3n–––objectsinwhichselect′n′objects,and,objects′3n′has′n′isidenticaland′2n′,differentobjects.Now,numberofways:n,identicalobjects–––––––––––––––––––––––:0,1,2.....3......nand,2n,differentobject–––––––––––––––––––––––:n,n−1,n−2,......n−3.......0thentheno.ofwaysandaddingtheno.ofways:2nc0+2nc1+2nc2+.......2nc3+......2ncn[weknowthat:2nc0+2nc1+2nc2+.....2ncn+.....2ncn+1....2nc2n=22n]Again,⇒2ncn+1+2ncn+2......2nc2n,wecanalsowriteas:⇒2nc2n−(n+1)+2nc2n(n+2).......2nc2n−2n|(ncr=ncn−r)⇒2ncn−1+2ncn−2.......2nc0Againwewrite:⇒2nc0+2nc1+2nc2+.....2ncn−1+2ncn+2ncn+1......2nc2n=22nadding2ncnintoR.H.S⇒2nc0+2nc1+2nc2+.....2ncn−1+2ncn––––––––––––––––––––––––––––––––––––––––+2ncn+1......2nc2n–––––––––––––––––––=22n+2ncnsupposewevaluedtheequn:s+s=22n+2ncn⇒2s=22n+2ncn⇒2s=22n+(2n)!n!n!∴s=22n−1+(2n)!2(n!)2(prove)

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