Find the indicated value:
a. 4C2
b. List all of the permutations of {a,b,c,d} when the elements are taken two at a time.
a. To find the value of 4C2, which represents "4 choose 2", we need to calculate the number of combinations of 4 items taken 2 at a time.
The formula for combinations is:
nCr = n! / (r!(n-r)!)
In this case, n is 4 and r is 2. So, substituting the values into the formula:
4C2 = 4! / (2!(4-2)!)
= 4! / (2! * 2!)
= (4 * 3 * 2!) / (2! * 2)
= (4 * 3) / 2
= 12 / 2
= 6
Therefore, 4C2 is equal to 6.
b. To list all of the permutations of {a, b, c, d} taken two at a time, we can use the permutation formula.
The formula for permutations is:
nP r = n! / (n - r)!
In this case, n is 4 and r is 2. So, substituting the values into the formula:
4P2 = 4! / (4 - 2)!
= 4! / 2!
= (4 * 3 * 2!) / 2!
= (4 * 3) / 2
= 12 / 2
= 6
Therefore, 4P2 is also equal to 6.
Now, to list all the permutations of {a, b, c, d} taken two at a time:
1. This can be done by taking the first element and pairing it with each of the remaining three elements. We obtain:
(a, b), (a, c), (a, d)
2. Next, we take the second element and pair it with the remaining two elements. We obtain:
(b, c), (b, d)
3. Finally, we take the third element and pair it with the last remaining element. We obtain:
(c, d)
So, the permutations of {a, b, c, d} taken two at a time are:
(a, b), (a, c), (a, d), (b, c), (b, d), (c, d)