Question

{"given":"The sum of coefficients in the expansion of $(a + b)^n$ is $4096$. We need to find the greatest coefficient in this binomial expansion.","key_observation":"The sum of coefficients in $(a + b)^n$ is obtained by substituting $a = 1$ and $b = 1$, giving $(1 + 1)^n = 2^n$. For a binomial expansion $(a + b)^n$ with even $n$, the greatest coefficient is the middle term coefficient $\\binom{n}{n/2}$.","option_analysis":[{"label":"(A)","text":"$1594$","verdict":"incorrect","explanation":"This value does not correspond to any standard binomial coefficient for $n = 12$. The calculation $\\binom{12}{6} = 924$ gives the correct maximum coefficient."},{"label":"(B)","text":"$792$","verdict":"incorrect","explanation":"This is equal to $\\binom{12}{5} = \\binom{12}{7} = 792$, which are the coefficients adjacent to the middle term, but not the maximum coefficient."},{"label":"(C)","text":"$924$","verdict":"correct","explanation":"Step 1: From $2^n = 4096 = 2^{12}$, we get $n = 12$.\nStep 2: For $(a + b)^{12}$, since $n = 12$ is even, the greatest coefficient is the middle term coefficient.\nStep 3: The middle term is $\\binom{12}{6}$ (the $7^{th}$ term).\nStep 4: Calculate $\\binom{12}{6} = \\frac{12!}{6! \\cdot 6!} = \\frac{12 \\times 11 \\times 10 \\times 9 \\times 8 \\times 7}{6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1}$.\nStep 5: Simplifying: $\\frac{665280}{720} = 924$."},{"label":"(D)","text":"$2924$","verdict":"incorrect","explanation":"This value is much larger than the correct answer and does not correspond to any binomial coefficient $\\binom{12}{r}$ for $0 \\leq r \\leq 12$."}],"answer":"(C)","formula_steps":[]}