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Find the highest power of 3 in ""^(50)C(...

Find the highest power of 3 in `""^(50)C_(10) and ""^(50)P_(10)`

Text Solution

Verified by Experts

We have
`""_(50)C_(10)=(50!)/(10!40!)`
`H_(3)(50!)=[(50)/(3)]+[(50)/(9)]+[(50)/(27)] +[(50)/(81)]=16+5+1+0=22`
`H_(3)(40!)=[(40)/(3)]+[(40)/(9)]+[(40)/(27)]+[(40)/(81)]=13+4+1+0=18`
`H_(3)(10!)=[(10)/(3)]+[(10)/(9)]+[(10)/(27)]`
`=3+1+0=4`
`therefore H_(3)(""^(50)C_(10))=H_(3)(50!)-H_(3)(40!)-H_(3)(10!)`
`=22-18-4=0`
Also `, H_(3)(""^(50)P_(10))=H_(3)(50!)-H_(3)(40!)=22-18=4`
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