What is the meaning of 1_C_1/2 = 2 in mathematics?

[member 141513]

is the answer of $$C^{1}_{1/2}=2$$?
is that meant:
we have 1 piece, what is the combination of 1-half?
thank you very much

 

[Nebuchadnezza]

No, it is C. Or it really deppends, if you integrate with respect to C, then clearly the answer is 1/2.

If C is a constant, and you integrate with respect to x, then your integral really says.

$$\int_{1/2}^{1} \, \text{dx} \, = \, \int_{x=1/2}^{x=1} \, \text{dx} \, = \, C|_{x=1/2}^{x=1} \, = \, C$$

Otherwise

$$\int_{1/2}^{1} 1 \, \text{dx} \, = \, \int_{x=1/2}^{x=1} 1 \, \text{dx} \, = \, x |_{x=1/2}^{x=1} \, = \, (1)-(\frac{1}{2}) \, = \, \frac{1}{2}$$

 

[member 141513]

No, it is C. Or it really deppends, if you integrate with respect to C, then clearly the answer is 1/2.

If C is a constant, and you integrate with respect to x, then your integral really says.

$$\int_{1/2}^{1} \, \text{dx} \, = \, \int_{x=1/2}^{x=1} \, \text{dx} \, = \, C|_{x=1/2}^{x=1} \, = \, C$$

Otherwise

$$\int_{1/2}^{1} 1 \, \text{dx} \, = \, \int_{x=1/2}^{x=1} 1 \, \text{dx} \, = \, x |_{x=1/2}^{x=1} \, = \, (1)-(\frac{1}{2}) \, = \, \frac{1}{2}$$

sorry sir , my question is about the combinatorial thing of 1 _C_ 1/2, the combination

 

[pwsnafu]

The combinatorics answer would be 2. You have 2 elements, each element is "one half part", and you choose 1 element.


However, this is not the answer to "1 choose 1/2" as in binomial coefficients. In analysis, "x choose y" for real x and y is given by the gamma function

$$\frac{\Gamma(x+1)}{\Gamma(y+1) \Gamma(x-y+1)}$$

which in this case (x=1, y=1/2) gives the answer 4 divided by pi. This is the answer given by http://www.wolframalpha.com/input/?i=Binomial[1,1/2.
Which answer being correct is dependent on context.