Question

If \( {}^{30}C_{30-r} + 3 \cdot {}^{30}C_{31-r} + 3 \cdot {}^{30}C_{32-r} + {}^{30}C_{33-r} = {}^{n}C_{r} \), then value of \( n \) is

Hint:

Whenever you see coefficients like \(1, 3, 3, 1\), think of the binomial expansion of \( (1+x)^3 \). It often helps combine several combination terms into one single combination.

Answer 1

Correct Answer: 33

Step 1: Rewrite the binomial coefficients in a convenient form.
Using the symmetry property of combinations, \[ {}^{30}C_{30-r} = {}^{30}C_{r}, \qquad {}^{30}C_{31-r} = {}^{30}C_{r-1}, \qquad {}^{30}C_{32-r} = {}^{30}C_{r-2}, \qquad {}^{30}C_{33-r} = {}^{30}C_{r-3}. \] So the given expression becomes \[ {}^{30}C_{r} + 3 \cdot {}^{30}C_{r-1} + 3 \cdot {}^{30}C_{r-2} + {}^{30}C_{r-3}. \]
Step 2: Identify the binomial pattern.
The coefficients \( 1, 3, 3, 1 \) are the binomial coefficients in the expansion of \[ (1+x)^3 = 1 + 3x + 3x^2 + x^3. \] Hence, this suggests applying the standard identity \[ {}^{n}C_{r} + 3 \cdot {}^{n}C_{r-1} + 3 \cdot {}^{n}C_{r-2} + {}^{n}C_{r-3} = {}^{n+3}C_{r}. \]
Step 3: Apply the identity.
Putting \( n = 30 \), we get \[ {}^{30}C_{r} + 3 \cdot {}^{30}C_{r-1} + 3 \cdot {}^{30}C_{r-2} + {}^{30}C_{r-3} = {}^{33}C_{r}. \]
Step 4: Compare with the given expression.
Since the given expression is equal to \[ {}^{n}C_{r}, \] we compare and obtain \[ {}^{n}C_{r} = {}^{33}C_{r}. \] Therefore, \[ n = 33. \]