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The value of (""^(47)C(4))/(""^(57)C(4))...

The value of `(""^(47)C_(4))/(""^(57)C_(4))+sum_(j=0)^(3)(""^(50-j)C_(3))/(""^(57)C_(53))+sum_(j=0)^(5)""^((56-k)C_(53-k))/(""^(57)C_(4))` is :

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We have, `.^(47)C_(4)+underset(j=0)overset(3)(sum).^(50-j)C_(3)+underset(k=0)overset(5)(sum).^(56-k)C_(53-k)`
`=.^(47)C_(4)+underset(j=0)overset(3)(sum).^(50-j)C_(3)+underset(k=0)overset(5)(.^(56)-k)C_(3)" "[because.^(n)C_(r)=.^(n)C_(n-r)]`
`=.^(47)C_(4)+(.^(50)C_(3)+.^(49)C_(3)+.^(48)C_(3)+.^(47)C_(3))`
`+(.^(56)C_(3)+.^(55)C_(3)+.^(54)C_(3)+.^(53)C_(3)+.^(52)C_(3)+.^(51)C_(3))`
`=.^(47)C_(4)+.^(47)C_(3)+.^(48)C_(3)+.^(49)C_(3)+.^(50)C_(3)+.^(51)C_(3)+.^(52)C_(3)+.^(52)C_(3)+.^(54)C_(3)+.^(55)C_(3)+.^(56)C_(3)`
`=(.^(47)C_(4)+.^(47)C_(3))+.^(48)C_(3)+.^(49)C_(3)+.^(50)C_(3)+.^(51)C_(3)+.^(52)C_(3)+.^(53)C_(3)+.^(54)C_(3)+.^(55)C_(3)+.^(56)C_(3)`
`=.^(48)C_(4)+.^(48)C_(3)+.^(49)C_(3)+.^(50)C_(3)+.^(51)C_(3)+ . . .+.^(56)C_(3)`
`=.^(49)C_(4)+.^(49)C_(3)+.^(50)C_(3)+ . . .+.^(56)C_(3)`
`=.^(56)C_(4)+.^(56)C_(3)=.^(57)C_(4)`
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