Answer:
360
Step-by-step explanation:
Here we are required to find [tex]_^{6}\textrm{P}_{4}[/tex]
It is a problem of Permutation and we must understand the formula for finding permutations.
The general formula for finding the permutation is given as below:
[tex]_^{m}\textrm{P}_{n}=\frac{m!}{(m-n)!}[/tex]
Hence
[tex]_^{6}\textrm{P}_{4}=\frac{6!}{(6-4)!}[/tex]
[tex]_^{6}\textrm{P}_{4}=\frac{6!}{2!}[/tex]
Where
[tex]m!=m\times(m-1)\times\(m-2)\times\cdot\cdot\cdot\codt\cdot3\times2\times1[/tex]
[tex]6!=6\times5\times4\times3\times2\times1[/tex]
[tex]2!=2\times1[/tex]
Hence
[tex]_^{6}\textrm{P}_{4}=\frac{6\times5\times4\times3\times2\times1}{2\times1}[/tex]
[tex]_^{6}\textrm{P}_{4}=6\times5\times4\times3[/tex]
[tex]_^{6}\textrm{P}_{4}=360[/tex]
360
A permutation is used to calculate how many ways to choose or know the various arrangements by considering the order.
The formula for finding the number of different ways or number of permutations of n different objects taken r at the time is
[tex]\boxed{ \ _nP_r \ or \ P(n, r) = \frac{n!}{(n - r)!} \ }[/tex]
Let us evaluate the value of ₆P₄.
[tex]\boxed{ \ _{6}P_4 \ or \ P(6, 4) = \frac{6!}{(6 - 4)!} \ }[/tex]
[tex]\boxed{ \ _{6}P_4 = \frac{6!}{2!} \ }[/tex]
Recall that [tex]\boxed{n! = n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1}[/tex] as n factorial.
[tex]\boxed{ \ _{6}P_4 = \frac{6 \times 5 \times 4 \times 3 \times 2!}{2!} \ }[/tex]
We expand 6! because there are 2! Inside it. Then we easily cross out 2! in the numerator and denominator.
[tex]\boxed{ \ _{6}P_4 = 6 \times 5 \times 4 \times 3 \ }[/tex]
[tex]\boxed{\boxed{ \ _{6}P_4 \ or \ P(6, 4) = 360 \ }}[/tex]
As a result, the expression ₆P₄ is 360.
Keywords: evaluate the expression 6P4, a permutation, how many ways, to choose, the various arrangements, by considering the order, the formula, finding the number of different ways, n different objects taken at that time, factorial, combination
