Question

Please solve the Problem given in the attachment. ​

Answer 1

Step-by-step explanation:

Question:-

Find the value of K if :-

$\rm ^{95} C_{4} + \sum \limits _ {j = 1} ^ {5} \: ^{100 - j} C_{3} = \: ^{100} C_{K}$

Formula used:-

• $\boxed{\rm ^{n} C_{r} + \: ^{n} C_{r + 1} = \: ^{n + 1} C_{r + 1}}$

Solution:-

$\rm ^{95} C_{4} + \sum \limits _ {j = 1} ^ {5} \: ^{100 - j} C_{3} = \: ^{100} C_{K}$

$\rm \implies^{95} C_{4} + \big( \: ^{100 - 1} C_{3} + \: ^{100 - 2} C_{3} + \: ^{100 - 3} C_{3} + \: ^{100 - 4} C_{3} + \: ^{100 - 5} C_{3} \big) = \: ^{100} C_{K}$

$\rm \implies ^{95} C_{4} + \big( \: ^{99} C_{3} + \: ^{98} C_{3} + \: ^{97} C_{3} + \: ^{96} C_{3} + \: ^{95} C_{3} \big) = \: ^{100} C_{K}$

$\rm \implies \big( \: ^{95} C_{4} + \: ^{95} C_{3} \big)+ \big( \: ^{99} C_{3} + \: ^{98} C_{3} + \: ^{97} C_{3} + \: ^{96} C_{3} \big) = \: ^{100} C_{K}$

$\rm \implies \big( \: ^{95 + 1} C_{3 + 1} \big)+ \big( \: ^{99} C_{3} + \: ^{98} C_{3} + \: ^{97} C_{3} + \: ^{96} C_{3} \big) = \: ^{100} C_{K}$

$\rm \implies \big( \: ^{96} C_{4}+ \: ^{96} C_{3} \big) + \: ^{97} C_{3} + \: ^{98} C_{3} + \: ^{99} C_{3} = \: ^{100} C_{K}$

$\rm \implies \big( \: ^{97} C_{4}+ \: ^{97} C_{3} \big) + \: ^{98} C_{3} + \: ^{99} C_{3} = \: ^{100} C_{K}$

$\rm \implies \big( \: ^{98} C_{4}+ \: ^{98} C_{3} \big) + \: ^{99} C_{3} = \: ^{100} C_{K}$

$\rm \implies \big( \: ^{99} C_{4}+ \: ^{99} C_{3} \big) = \: ^{100} C_{K}$

$\rm \implies \: ^{100} C_{4}= \: ^{100} C_{K}$

$\rm \implies 4= K$

$\longrightarrow \underline{ \boxed{\rm \therefore \: \: K = 4}}$