Evaluate the following expression:
[tex]\[ \frac{1}{2} + \frac{2}{3} - \binom{31}{12}^0 + \frac{9}{4} \][/tex]



Answer :

Let's solve the given expression step by step:

Expression: [tex]\(\frac{1}{2} + \frac{2}{3} - \binom{31}{12}^0 + \frac{9}{4}\)[/tex]

1. Identify and simplify the binomial coefficient term raised to the power of 0:
[tex]\(\binom{31}{12}^0\)[/tex]

Any term raised to the power of 0 equals 1:
[tex]\(\binom{31}{12}^0 = 1\)[/tex]

2. Substitute this value into the original expression:
[tex]\(\frac{1}{2} + \frac{2}{3} - 1 + \frac{9}{4}\)[/tex]

3. Convert each fraction to have a common denominator for addition and subtraction:

- [tex]\(\frac{1}{2} = \frac{2}{4}\)[/tex]
- [tex]\(\frac{2}{3} = \frac{8}{12}\)[/tex]
- [tex]\(\frac{9}{4} = \frac{27}{12}\)[/tex]

However, let's use decimals to make calculation simpler.

4. Convert fractions to decimal:
- [tex]\(\frac{1}{2} = 0.5\)[/tex]
- [tex]\(\frac{2}{3} \approx 0.6666666666666666\)[/tex]
- [tex]\(\frac{9}{4} = 2.25\)[/tex]

5. Calculate the sum and subtraction:
[tex]\[ 0.5 + 0.6666666666666666 - 1 + 2.25 \][/tex]

6. Perform each operation sequentially:

- Add [tex]\(0.5\)[/tex] and [tex]\(0.6666666666666666\)[/tex]:
[tex]\[ 0.5 + 0.6666666666666666 \approx 1.1666666666666665 \][/tex]

- Subtract [tex]\(1\)[/tex] from the result:
[tex]\[ 1.1666666666666665 - 1 = 0.16666666666666652 \][/tex]

- Finally, add [tex]\(2.25\)[/tex]:
[tex]\[ 0.16666666666666652 + 2.25 = 2.4166666666666665 \][/tex]

So, the result of the expression [tex]\(\frac{1}{2} + \frac{2}{3} - \binom{31}{12}^0 + \frac{9}{4}\)[/tex] is [tex]\(2.4166666666666665\)[/tex].