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}

• A. 9 or 10
• B. 7 or 12
• C. 8 or 10
• D. 8 or 11

Hint:

Always start the sum from the smallest $n$ values to trigger the chain reaction of Pascal's identity.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We use the identity \( ^nC_r + ^nC_{r-1} = ^{n+1}C_r \) repeatedly to simplify the sum.
Step 2: Key Formula or Approach:
Pascal's identity: \( ^nC_r + ^nC_{r-1} = ^{n+1}C_r \).
Step 3: Detailed Explanation:
Starting with the first two terms:
\( ^{15}C_4 + ^{15}C_5 = ^{16}C_5 \).
Add the next term:
\( ^{16}C_5 + ^{16}C_6 = ^{17}C_6 \).
Add the next term:
\( ^{17}C_6 + ^{17}C_7 = ^{18}C_7 \).
Add the final term:
\( ^{18}C_7 + ^{18}C_8 = ^{19}C_8 \).
We are given \( ^{19}C_r = ^{19}C_8 \).
By symmetry, \( ^nC_r = ^nC_{n-r} \).
So \( r = 8 \) or \( r = 19 - 8 = 11 \).
Step 4: Final Answer:
The values of \( r \) are 8 or 11.