If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
Solving the Combination Product Equation for Alpha
The problem requires finding the value of $30\alpha$ based on an equation involving a product of sums of reciprocal binomial coefficients.
Simplifying the General Term
Let the general term in the product be $T_k = \left(\frac{1}{^{15}C_k} + \frac{1}{^{15}C_{k+1}}\right)$ for $k = 0, 1, \dots, 12$.
We use the identity: $ \frac{1}{^{n}C_k} + \frac{1}{^{n}C_{k+1}} = \frac{k!(n-k)!}{n!} + \frac{(k+1)!(n-k-1)!}{n!} $ $ = \frac{k!(n-k-1)! [ (n-k) + (k+1) ]}{n!} = \frac{(n+1) k!(n-k-1)!}{n!} $
Applying this with $n=15$, the general term becomes:
$ T_k = \frac{(15+1) k!(15-k-1)!}{15!} = \frac{16 \cdot k! (14-k)!}{15!} $
Calculating the Product (LHS)
The left-hand side (LHS) of the equation is the product $ P = \prod_{k=0}^{12} T_k $.
$ P = \prod_{k=0}^{12} \left( \frac{16 \cdot k! (14-k)!}{15!} \right) $
$ P = \left( \frac{16}{15!} \right)^{13} \prod_{k=0}^{12} [k! (14-k)!] $
Let $ \text{ProdFact} = \prod_{k=0}^{12} [k! (14-k)!] $. Then, $ P = \left( \frac{16}{15!} \right)^{13} \text{ProdFact} $.
Analyzing the Right-Hand Side (RHS)
The right-hand side (RHS) is given as $ \frac{\alpha^{13}}{D} $, where $ D = ^{14}C_0 \cdot ^{14}C_1 \dots ^{14}C_{12} $.
$ D = \prod_{k=0}^{12} ^{14}C_k = \prod_{k=0}^{12} \frac{14!}{k!(14-k)!} $
$ D = \frac{14!^{13}}{\prod_{k=0}^{12} k! (14-k)!} = \frac{14!^{13}}{\text{ProdFact}} $
Equating and Solving for Alpha
Equating the LHS ($P$) and RHS:
$ \left( \frac{16}{15!} \right)^{13} \text{ProdFact} = \frac{\alpha^{13}}{D} $
Substitute the expression for $D$:
$ \left( \frac{16}{15!} \right)^{13} \text{ProdFact} = \frac{\alpha^{13} \cdot \text{ProdFact}}{14!^{13}} $
Cancel $ \text{ProdFact} $ from both sides:
$ \left( \frac{16}{15!} \right)^{13} = \frac{\alpha^{13}}{14!^{13}} $
Take the 13th root of both sides:
$ \frac{16}{15!} = \frac{\alpha}{14!} $
Solve for $ \alpha $:
$ \alpha = \frac{16 \cdot 14!}{15!} = \frac{16 \cdot 14!}{15 \cdot 14!} = \frac{16}{15} $
Calculating the Final Value
The question asks for the value of $30\alpha$.
$ 30\alpha = 30 \times \frac{16}{15} $
$ 30\alpha = 2 \times 16 = 32 $
Thus, the value of $30\alpha$ is 32.
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
- If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\frac{1}{x^3} - x^4\right)^n$, $x \neq 0$, is zero, then the value of $n$ is _________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
