Let's evaluate the binomial coefficients given in the problem. We'll use the generalized binomial coefficient formula for any real number [tex]\( n \)[/tex] and integer [tex]\( k \)[/tex]:
[tex]\[ \binom{n}{k} = \frac{n(n-1)(n-2) \cdots (n-k+1)}{k!} \][/tex]
### Part (i):
Evaluate [tex]\(\binom{-2}{4}\)[/tex]:
[tex]\[ \binom{-2}{4} = \frac{(-2)(-3)(-4)(-5)}{4!} \][/tex]
First, compute the numerator:
[tex]\[ (-2)(-3)(-4)(-5) = 120 \][/tex]
Next, compute the denominator ([tex]\(4!\)[/tex]):
[tex]\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \][/tex]
So,
[tex]\[ \binom{-2}{4} = \frac{120}{24} = 5 \][/tex]
### Part (ii):
Evaluate [tex]\(\binom{-\frac{1}{4}}{3}\)[/tex]:
[tex]\[ \binom{-\frac{1}{4}}{3} = \frac{\left( -\frac{1}{4} \right) \left( -\frac{1}{4} - 1 \right) \left( -\frac{1}{4} - 2 \right)}{3!} \][/tex]
First, compute the terms in the numerator:
[tex]\[ -\frac{1}{4} \][/tex]
[tex]\[ -\frac{1}{4} - 1 = -\frac{1}{4} - \frac{4}{4} = -\frac{5}{4} \][/tex]
[tex]\[ -\frac{1}{4} - 2 = -\frac{1}{4} - \frac{8}{4} = -\frac{9}{4} \][/tex]
So the numerator is:
[tex]\[ \left( -\frac{1}{4} \right) \left( -\frac{5}{4} \right) \left( -\frac{9}{4} \right) = -\frac{1}{4} \times -\frac{5}{4} \times -\frac{9}{4} = -\frac{1 \times 5 \times 9}{4 \times 4 \times 4} = -\frac{45}{64} \][/tex]
Next, compute the denominator ([tex]\(3!\)[/tex]):
[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]
So,
[tex]\[ \binom{-\frac{1}{4}}{3} = \frac{-\frac{45}{64}}{6} = -\frac{45}{64 \times 6} = -\frac{45}{384} = -\frac{15}{128} \][/tex]
### Summary:
(i) [tex]\(\binom{-2}{4} = 5\)[/tex]
(ii) [tex]\(\binom{-\frac{1}{4}}{3} = -\frac{15}{128}\)[/tex]
