Question

# Find the number of ways in which : (a) a selection (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'.

Solution

## Given word PROPORTION$$P-2,R-2,0-3,T,I,N,$$$$(i)$$ Words with all distinct letters $$= 6_{c4}=15$$waysArrangement=$$6_{p4}=360$$ ways$$(ii)$$ One letter repeat twice Selection $$=3_{c1}\times 5_{c2}$${i.e one from $$P/R/O$$ & $$2$$ from the $$5$$} Arrangement $$=3_{c1}\times 5_{c2}\times \dfrac{4!}{2!}=360$$ $$(iii)$$Two letter repeated twice Selection $$=3_{c2}=3$$ [i.e two from $$P/R/O$$]Arrangement $$=3_{c2}\times \dfrac{4!}{2!.2!}=3\times 6=18$$$$(iv)$$ Three letter some Selection $$=1_{c1}\times 5_{c1}=5$$ [i.e one from orther $$3$$ all $$'0"$$  ]Arrangement$$=5\times\dfrac{4!}{3!}=20$$ $$\therefore$$ Total selection $$= 15+30+3+5$$$$=53$$ waysTotal arrangement $$=360+360+18+20$$$$=758$$ waysMathematics

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