The value of the expression { }^{47} C_4 + su | vedclass

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(C) We use the identity ${ }^{n} C_{r} + { }^{n} C_{r-1} = { }^{n+1} C_{r}$.Expanding the summation, the expression is:${ }^{47} C_4 + { }^{51} C_3 +

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${ }^{52} C_4$

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(C) We use the identity ${ }^{n} C_{r} + { }^{n} C_{r-1} = { }^{n+1} C_{r}$.Expanding the summation, the expression is:${ }^{47} C_4 + { }^{51} C_3 + { }^{50} C_3 + { }^{49} C_3 + { }^{48} C_3 + { }^{47} C_3$Using the identity ${ }^{47} C_4 + { }^{47} C_3 = { }^{48} C_4$, the expression becomes:${ }^{48} C_4 + { }^{48} C_3 + { }^{49} C_3 + { }^{50} C_3 + { }^{51} C_3$Applying the identity repeatedly:${ }^{48} C_4 + { }^{48} C_3 = { }^{49} C_4$${ }^{49} C_4 + { }^{49} C_3 = { }^{50} C_4$${ }^{50} C_4 + { }^{50} C_3 = { }^{51} C_4$${ }^{51} C_4 + { }^{51} C_3 = { }^{52} C_4$Thus, the final value is ${ }^{52} C_4$.

The value of the expression ${ }^{47} C_4 + \sum_{j=1}^5 { }^{52-j} C_3$ is

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