Key Concepts and Formulas

• Binomial Coefficient Definition: ${n \choose r} = \frac{n!}{r!(n-r)!}$
• Binomial Coefficient Ratio: ${n \choose r} = \frac{n}{r} {n-1 \choose r-1}$
• Solving simultaneous equations.

Step-by-Step Solution

Step 1: Express the given ratios using the binomial coefficient formula.

We are given the ratio ${ }^{n+1} C_{r+1}:{ }^n C_r:{ }^{n-1} C_{r-1}=55: 35: 21$. We need to use this information to find the values of $n$ and $r$.

Step 2: Rewrite the ratios as fractions.

We can rewrite the given ratios as: $\frac{{n+1 \choose r+1}}{{n \choose r}} = \frac{55}{35} = \frac{11}{7}$ and $\frac{{n \choose r}}{{n-1 \choose r-1}} = \frac{35}{21} = \frac{5}{3}$

Step 3: Simplify the binomial coefficient ratios using the formula.

Using the formula ${n \choose r} = \frac{n}{r} {n-1 \choose r-1}$, we can rewrite the ratios as: $\frac{{n+1 \choose r+1}}{{n \choose r}} = \frac{\frac{(n+1)!}{(r+1)!(n-r)!}}{\frac{n!}{r!(n-r)!}} = \frac{(n+1)! r!}{(r+1)! n!} = \frac{n+1}{r+1}$ and $\frac{{n \choose r}}{{n-1 \choose r-1}} = \frac{\frac{n!}{r!(n-r)!}}{\frac{(n-1)!}{(r-1)!(n-r)!}} = \frac{n! (r-1)!}{r! (n-1)!} = \frac{n}{r}$

Step 4: Establish equations based on the simplified ratios.

From Step 2 and Step 3, we have the following equations: $\frac{n+1}{r+1} = \frac{11}{7}$ $\frac{n}{r} = \frac{5}{3}$ These can be rewritten as: $7(n+1) = 11(r+1) \implies 7n + 7 = 11r + 11 \implies 7n - 11r = 4 \quad \text{.... (1)}$ $3n = 5r \implies 3n - 5r = 0 \quad \text{.... (2)}$

Step 5: Solve the system of linear equations.

We have a system of two linear equations with two variables, $n$ and $r$: $7n - 11r = 4 \quad \text{.... (1)}$ $3n - 5r = 0 \quad \text{.... (2)}$ From equation (2), we can express $n$ in terms of $r$: $n = \frac{5r}{3}$ Substitute this into equation (1): $7\left(\frac{5r}{3}\right) - 11r = 4$ $\frac{35r}{3} - 11r = 4$ $\frac{35r - 33r}{3} = 4$ $\frac{2r}{3} = 4$ $2r = 12$ $r = 6$ Now, substitute $r = 6$ back into the equation for $n$: $n = \frac{5(6)}{3} = \frac{30}{3} = 10$

Step 6: Calculate the value of 2n + 5r.

We have $n = 10$ and $r = 6$. Therefore, $2n + 5r = 2(10) + 5(6) = 20 + 30 = 50$

Common Mistakes & Tips

• Be careful when simplifying the ratios of binomial coefficients. Ensure you correctly apply the formulas.
• Double-check your algebraic manipulations when solving the system of equations to avoid errors.
• Remember to express the final answer in the form requested by the problem (in this case, $2n + 5r$).

Summary

We were given a ratio of binomial coefficients and asked to find the value of $2n + 5r$. By expressing the ratios as fractions, simplifying them using the binomial coefficient formula, and solving the resulting system of linear equations, we found that $n = 10$ and $r = 6$. Therefore, $2n + 5r = 50$.

Final Answer

The final answer is \boxed{50}, which corresponds to option (D).