Questions: Evaluate 10 mathrmC6 and 9 mathrmP3. 10 mathrmC6=square 9 mathrmP3=square

Solution Steps

To evaluate the combination \({ }_{10} \mathrm{C}_{6}\), we use the formula for combinations: \({ }_{n} \mathrm{C}_{r} = \frac{n!}{r!(n-r)!}\). For the permutation \({ }_{9} \mathrm{P}_{3}\), we use the formula for permutations: \({ }_{n} \mathrm{P}_{r} = \frac{n!}{(n-r)!}\). We will calculate these values using Python's math library, which provides a factorial function.

To find the value of \( { }_{10} \mathrm{C}_{6} \), we use the combination formula: \[ { }_{n} \mathrm{C}_{r} = \frac{n!}{r!(n-r)!} \] Substituting \( n = 10 \) and \( r = 6 \): \[ { }_{10} \mathrm{C}_{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6! \cdot 4!} \] Calculating this gives: \[ { }_{10} \mathrm{C}_{6} = 210 \]

To find the value of \( { }_{9} \mathrm{P}_{3} \), we use the permutation formula: \[ { }_{n} \mathrm{P}_{r} = \frac{n!}{(n-r)!} \] Substituting \( n = 9 \) and \( r = 3 \): \[ { }_{9} \mathrm{P}_{3} = \frac{9!}{(9-3)!} = \frac{9!}{6!} \] Calculating this gives: \[ { }_{9} \mathrm{P}_{3} = 504 \]

Final Answer