Key Concepts and Formulas

• Combinations with Repetition (Stars and Bars): The number of ways to choose $k$ items from $n$ distinct types with repetition allowed is ${}^{n+k-1}{C_k} = {}^{n+k-1}{C_{n-1}}$.
• Permutations with Repetition: The number of distinct arrangements of $N$ objects, where $n_1$ are of one type, $n_2$ are of another type, and so on, is given by $\frac{N!}{n_1! n_2! \dots}$.
• Binomial Coefficient Identity: ${}^{n}{C_r} = {}^{n}{C_{n-r}}$.

Step-by-Step Solution

Step 1: Analyze Statement - 1 and apply the combinations with repetition formula.

• What & Why: We need to determine if the calculation of the number of ways to buy six ice creams of five types with repetition allowed, as stated in Statement - 1, is correct. We will use the Stars and Bars formula.
• Math: We have $n = 5$ (number of ice cream types) and $k = 6$ (number of ice creams bought). Applying the formula for combinations with repetition: ${}^{n+k-1}{C_k} = {}^{5+6-1}{C_6} = {}^{10}{C_6}$ Using the identity ${}^{n}{C_r} = {}^{n}{C_{n-r}}$, we have: ${}^{10}{C_6} = {}^{10}{C_{10-6}} = {}^{10}{C_4}$
• Reasoning: The formula gives us ${}^{10}{C_6}$ or ${}^{10}{C_4}$. Statement - 1 claims the answer is ${}^{10}{C_5}$. Since ${}^{10}{C_4} \neq {}^{10}{C_5}$, Statement - 1 is incorrect.

Step 2: Conclude about Statement - 1.

• What & Why: Based on our calculation in Step 1, we determine the truth value of Statement - 1.
• Reasoning: Since our calculation resulted in ${}^{10}{C_4}$ (or ${}^{10}{C_6}$) and the statement claims ${}^{10}{C_5}$, Statement - 1 is false.

Step 3: Analyze Statement - 2 and relate it to permutations with repetition.

• What & Why: We need to determine if the number of ways to buy six ice creams is equal to the number of ways to arrange 6 A's and 4 B's. This involves permutations with repetition.
• Math: The number of ways to arrange 6 A's and 4 B's is given by: $\frac{10!}{6!4!} = {}^{10}{C_6} = {}^{10}{C_4}$
• Reasoning: The expression $\frac{10!}{6!4!}$ is equivalent to the binomial coefficient ${}^{10}{C_6}$ (or ${}^{10}{C_4}$). This is the same result we obtained in Step 1 for the number of ways to buy the ice creams. The 'A's represent the chosen ice creams, and the 'B's represent dividers between the different types.

Step 4: Explain the Stars and Bars connection to A's and B's.

• What & Why: We need to explain why the A's and B's arrangement is mathematically equivalent to the Stars and Bars representation.
• Reasoning: In the Stars and Bars method, we have 6 stars (ice creams) and 4 bars (dividers between the 5 types). Arranging these 6 stars and 4 bars is the same as choosing 6 positions for the stars (or 4 positions for the bars) out of a total of 10 positions. This is represented by ${}^{10}{C_6}$ (or ${}^{10}{C_4}$). This is exactly equivalent to arranging 6 A's and 4 B's, where A's represent the stars and B's represent the bars.

Step 5: Conclude about Statement - 2.

• What & Why: Based on our calculations and reasoning, we determine the truth value of Statement - 2.
• Reasoning: The number of ways to arrange 6 A's and 4 B's is ${}^{10}{C_4}$, which is the same as the number of ways to choose the ice creams. Therefore, Statement - 2 is true.

Step 6: Select the correct option.

• What & Why: Since Statement - 1 is false and Statement - 2 is true, we need to choose the option that reflects this.
• Reasoning: Option (A) states "Statement - 1 is false, Statement - 2 is true". This matches our findings.

Common Mistakes & Tips

• Confusing n and k: Ensure you correctly identify 'n' as the number of types (ice cream flavors) and 'k' as the number of items chosen (ice creams bought).
• Using the Wrong Formula: Avoid using combinations without repetition (${}^{n}{C_k}$) when repetition is allowed. Use the Stars and Bars formula ${}^{n+k-1}{C_k}$.
• Forgetting the Binomial Identity: Remember that ${}^{N}{C_R} = {}^{N}{C_{N-R}}$. This can simplify calculations and make connections clearer.

Summary

Statement - 1 incorrectly calculates the number of ways to choose the ice creams. Statement - 2 correctly equates the number of ways to choose the ice creams to the number of ways to arrange 6 A's and 4 B's. Therefore, Statement - 1 is false and Statement - 2 is true, which corresponds to option (A).

Final Answer The final answer is $\boxed{A}$.