Question

The value of ${}^4^9C_3 + {}^4^8C_3 + {}^4^7C_3 + {}^4^6C_3 + {}^4^5C_3 + {}^4^5C_4$ is:

• A. ${}^5^0C_4$
• B. ${}^5^0C_3$
• C. ${}^5^0C_2$
• D. ${}^5^0C_1}$

Answer 1

The Correct Option is A

Using the property of binomial coefficients, we know the recursive addition property: ${}^{n+1}C_r = {}^nC_r + {}^nC_{r-1}$. Applying this property iteratively, we can combine the terms as follows: \[ {}^{49}C_3 + {}^{48}C_3 + {}^{47}C_3 + {}^{46}C_3 + {}^{45}C_3 + {}^{45}C_4 = {}^{50}C_4 \] This sum represents the binomial coefficient ${}^{50}C_4$, as derived from the addition property of binomial coefficients.