Question:
If \( ^{15}C_4 + ^{15}C_5 + ^{16}C_6 + ^{17}C_7 + ^{18}C_8 = ^{19}C_r \), then the value of \( r \) is equal to}
If \( ^{15}C_4 + ^{15}C_5 + ^{16}C_6 + ^{17}C_7 + ^{18}C_8 = ^{19}C_r \), then the value of \( r \) is equal to}
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Always start the sum from the smallest $n$ values to trigger the chain reaction of Pascal's identity.
Updated On: Apr 30, 2026
- 9 or 10
- 7 or 12
- 8 or 10
- 8 or 11
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The Correct Option is D
Solution and Explanation
Step 1: Pascal's Identity
$^nC_r + ^nC_{r-1} = ^{n+1}C_r$.
Step 2: Simplify Sum
- $^{15}C_5 + ^{15}C_4 = ^{16}C_5$.
- $^{16}C_6 + ^{16}C_5 = ^{17}C_6$.
- $^{17}C_7 + ^{17}C_6 = ^{18}C_7$.
- $^{18}C_8 + ^{18}C_7 = ^{19}C_8$.
Step 3: Use Complementary Property
$^{19}C_8 = ^{19}C_{19-8} = ^{19}C_{11}$.
Step 4: Conclusion
$r$ can be 8 or 11.
Final Answer:(D)
$^nC_r + ^nC_{r-1} = ^{n+1}C_r$.
Step 2: Simplify Sum
- $^{15}C_5 + ^{15}C_4 = ^{16}C_5$.
- $^{16}C_6 + ^{16}C_5 = ^{17}C_6$.
- $^{17}C_7 + ^{17}C_6 = ^{18}C_7$.
- $^{18}C_8 + ^{18}C_7 = ^{19}C_8$.
Step 3: Use Complementary Property
$^{19}C_8 = ^{19}C_{19-8} = ^{19}C_{11}$.
Step 4: Conclusion
$r$ can be 8 or 11.
Final Answer:(D)
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