If left(frac1^15C_0 + frac1^15C_1right) left(frac1^15C_1 + frac1^15C_2right) dots left(frac1^15C_12 + frac1^15C_13right) = fracalpha^13^14C_0 cdot ^14C_1 dots ^14C_12$\left(\frac{1}{^{15}C_{0}} + \frac{1}{^{15}C_{1}}\right) \left(\frac{1}{^{15}C_{1}} + \frac{1}{^{15}C_{2}}\right) \dots \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right) = \frac{\alpha^{13}}{^{14}C_{0} \cdot ^{14}C_{1} \dots ^{14}C_{12}}$, then 30alpha$30\alpha$ is equal to ____.

### Related Formula ^nC_r + ^nC_r+1 = ^n+1C_r+1$$^nC_r + ^nC_{r+1} = ^{n+1}C_{r+1}$$ frac^n+1C_r+1n+1 = frac^nC_rr+1$$\frac{^{n+1}C_{r+1}}{n+1} = \frac{^nC_r}{r+1}$$ ### Core Logic Simplify the general term of the product: T_r = frac1^15C_r + frac1^15C_r+1 = frac^15C_r+1 + ^15C_r^15C_r cdot ^15C_r+1 = frac^16C_r+1^15C_r cdot ^15C_r+1$$T_r = \frac{1}{^{15}C_r} + \frac{1}{^{15}C_{r+1}} = \frac{^{15}C_{r+1} + ^{15}C_r}{^{15}C_r \cdot ^{15}C_{r+1}} = \frac{^{16}C_{r+1}}{^{15}C_r \cdot ^{15}C_{r+1}}$$ ### Step 1: Simplify General Term Using the relation ^16C_r+1 = frac16r+1 ^15C_r$^{16}C_{r+1} = \frac{16}{r+1} {^{15}C_r}$: T_r = fracfrac16r+1 ^15C_r^15C_r cdot ^15C_r+1 = frac16(r+1) cdot ^15C_r+1$$T_r = \frac{\frac{16}{r+1} {^{15}C_r}}{^{15}C_r \cdot ^{15}C_{r+1}} = \frac{16}{(r+1) \cdot ^{15}C_{r+1}}$$ Also, (r+1) cdot ^15C_r+1 = 15 cdot ^14C_r$(r+1) \cdot ^{15}C_{r+1} = 15 \cdot ^{14}C_r$. T_r = frac1615 cdot ^14C_r = frac16/15^14C_r$$T_r = \frac{16}{15 \cdot ^{14}C_r} = \frac{16/15}{^{14}C_r}$$ ### Step 2: Take the Product We need the product from r = 0$r = 0$ to 12$12$: prod_r=0^12 T_r = prod_r=0^12 frac16/15^14C_r = frac(16/15)^13prod_r=0^12 ^14C_r$$\prod_{r=0}^{12} T_r = \prod_{r=0}^{12} \frac{16/15}{^{14}C_r} = \frac{(16/15)^{13}}{\prod_{r=0}^{12} {^{14}C_r}}$$ = fracleft(frac1615right)^13^14C_0 cdot ^14C_1 dots ^14C_12$$= \frac{\left(\frac{16}{15}\right)^{13}}{^{14}C_0 \cdot ^{14}C_1 \dots ^{14}C_{12}}$$ ### Step 3: Evaluate alpha Comparing with the given RHS fracalpha^13textproduct$\frac{\alpha^{13}}{\text{product}}$: alpha = frac1615$$\alpha = \frac{16}{15}$$ Then 30alpha = 30 left( frac1615 right) = 32$30\alpha = 30 \left( \frac{16}{15} \right) = 32$. ### Pattern Recognition Product series of binomial fractions usually collapse by pairing the sum into Pascal's identity, separating the n$n$ coefficient and canceling factorials vertically. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Binomial Theorem