Determine whether the statement below is true or false. If it is false, rewrite it as a true statement. \[ { }_{7} \mathrm{C}_{5}={ }_{7} \mathrm{C}_{2} \] Choose the correct answer below. A. This statement is false. A true statement is ${ }_{2} \mathrm{C}_{7}={ }_{5} \mathrm{C}_{7}$. B. This statement is false. A true statement is ${ }_{7} \mathrm{C}_{5}={ }_{7} \mathrm{P}_{2}$. C. This statement is true. D. This statement is false. A true statement is $\mathrm{C}_{4}={ }_{7} \mathrm{C}_{2}$
Solution
Step 1 ：The problem is asking whether the number of ways to choose 5 items from 7 is the same as the number of ways to choose 2 items from 7. This is a property of combinations, where choosing r items from n is the same as choosing (n-r) items from n.
Step 2 ：To verify this, we need to calculate both sides of the equation and compare them.
Step 3 ：Using the combination formula \( { }_{n} \mathrm{C}_{r} = \frac{n!}{r!(n-r)!} \), we can calculate the left side of the equation as \( { }_{7} \mathrm{C}_{5} = 21.0 \).
Step 4 ：Similarly, we can calculate the right side of the equation as \( { }_{7} \mathrm{C}_{2} = 21.0 \).
Step 5 ：Since both sides of the equation are equal, the statement is true.
Step 6 ：Final Answer: \(\boxed{{ }_{7} \mathrm{C}_{5}={ }_{7} \mathrm{C}_{2}}\) is a true statement.
