Question

Calculate the following: a. \( { }_{5} P_{3} \) e. \( { }_{12} P_{3} \) b. \( { }_{8} P_{5} \) f. \( { }_{5} C_{3} \) c. \( { }_{6} P_{6} \) g. 8 ! d. \( 2 ! 3 \) ! h. \( 6 ! / 2 \) !

Answer 1

a.
`\( { }_(5) P_(3) = 60 \)`

b.
`\( { }_(8) P_(5) = 6720 \)`

c.
`\( { }_(6) P_(6) = 720 \)`

d.
`\( 2 ! 3 ! = 12 \)`

e.
`\( { }_(12) P_(3) = 1320 \)`

f.
`\( { }_(5) C_(3) = 10 \)`

g.
`\( 8 ! = 40320 \)`

h.
`\( (6 !)/(2 !) = 360 \)`

Step-by-step explanation:

a.
`\( { }_(5) P_(3) \)` represents the permutation of 3 elements chosen from a set of 5. The formula for permutations is
`\( nPr = (n!)/((n-r)!) \), so \( { }_(5) P_(3) = (5!)/((5-3)!) = (5!)/(2!) = 60 \).`

b.
`\( { }_(8) P_(5) \)` is the permutation of 5 elements chosen from a set of 8, calculated using the permutation formula.
`\( { }_(8) P_(5) = (8!)/((8-5)!) = 6720 \).`

c.
`\( { }_(6) P_(6) \)` is the permutation of all 6 elements in a set of 6, giving
`\( 6! = 720 \).`

d.
`\( 2 ! 3 ! \)` is the factorial of 2 multiplied by the factorial of 3, which equals 12.

e.
`\( { }_(12) P_(3) \)` represents the permutation of 3 elements chosen from a set of 12. Calculated using the permutation formula,
`\( { }_(12) P_(3) = (12!)/((12-3)!) = 1320 \).`

f.
`\( { }_(5) C_(3) \)` is the combination of 3 elements chosen from a set of 5. The combination formula is
`\( nCr = (n!)/(r!(n-r)!) \), so \( { }_(5) C_(3) = (5!)/(3!(5-3)!) = 10 \).`

g.
`\( 8 ! \)` is the factorial of 8, which is
`\( 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40320 \).`

h.
`\( (6 !)/(2 !) \)`is the factorial of 6 divided by the factorial of 2, giving
`\( (6!)/(2!) = 360 \).`