KCET · Maths · Permutation Combination
The value of
\({ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4\) is
- A \({ }^{50} C_4\)
- B \({ }^{50} C_3\)
- C \({ }^{50} C_2\)
- D \({ }^{50} C_1\)
Answer & Solution
Correct Answer
(A) \({ }^{50} C_4\)
Step-by-step Solution
Detailed explanation
\(\because{ }^n C_r+{ }^n C_{r+1}={ }^{n+1} C_{r+1}\)
Given, \({ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4\)
\(={ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{46} C_4\)
\(={ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{47} C_4\)
\(={ }^{49} C_3+{ }^{48} C_3+{ }^{48} C_4\)
\(={ }^{49} C_3+{ }^{49} C_4={ }^{50} C_4\)
Given, \({ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4\)
\(={ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{46} C_4\)
\(={ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{47} C_4\)
\(={ }^{49} C_3+{ }^{48} C_3+{ }^{48} C_4\)
\(={ }^{49} C_3+{ }^{49} C_4={ }^{50} C_4\)
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