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Question: Find the value of \(^{10}C_{4}+^{9}C_{4}+^{8}C_{4}+…+^{5}C_{4}\) (a) \(^{11}C_{5}\) (b) \(^{11}C...

Find the value of 10C4+9C4+8C4+…+5C4^{10}C_{4}+^{9}C_{4}+^{8}C_{4}+…+^{5}C_{4}
(a) 11C5^{11}C_{5}
(b) 11C4^{11}C_{4}
(c ) 11C7^{11}C_{7}
(d) 11C5−1^{11}C_{5}-1

Explanation

Solution

In these types of question we need to use the property of combination which is nCr+nCr−1=n+1Cr^{n}C_{r}+^{n}C_{r-1}=^{n+1}C_{r}

Complete step-by-step answer:
Given, the series 10C4+9C4+8C4+…+5C4^{10}C_{4}+^{9}C_{4}+^{8}C_{4}+…+^{5}C_{4}
Add 5C5^{5}C_{5} and subtract 1 from the series to keep the series same
=10C4+9C4+8C4+7C4+6C4+5C4+5C5−1^{10}C_{4}+^{9}C_{4}+^{8}C_{4}+^{7}C_{4}+^{6}C_{4}+^{5}C_{4}+^{5}C_{5}-1
Use the property of combination,
=10C4+9C4+8C4+7C4+6C4+6C5−1^{10}C_{4}+^{9}C_{4}+^{8}C_{4}+^{7}C_{4}+^{6}C_{4}+^{6}C_{5}-1
Again take the fifth and sixth term together and use the property of combination,
=10C4+9C4+8C4+7C4+7C5−1^{10}C_{4}+^{9}C_{4}+^{8}C_{4}+^{7}C_{4}+^{7}C_{5}-1

Similarly take the fourth and fifth term together and use the property of combination,
=10C4+9C4+8C4+8C5−1^{10}C_{4}+^{9}C_{4}+^{8}C_{4}+^{8}C_{5}-1

Again take the third and fourth term together and use the property of combination,
=10C4+9C4+9C5−1^{10}C_{4}+^{9}C_{4}+^{9}C_{5}-1

Take the second and third term together and use the property of combination,
=10C4+10C5−1^{10}C_{4}+^{10}C_{5}-1

Finally take the first 2 terms to reduce the series,
=11C5−1^{11}C_{5}-1

Note: In such types of questions we need to add some value as a solution requirement and subtract 1 in such a way so that the series remains the same and we get to use the property of combination.