**Solve the following. (35 pts) 1. How many 3-digit number can be formed from digits 1 – 5 if: a. If repetition is allowed? (3 pts) b. If repetition is not allowed? (3 pts) 2. How many strings of length 4 can be formed using the letters COMPUTER if repetitions are not allowed? (3 pts) 3. Out of 17 regions in the country, ten will be chosen to be included in a survey. How many ways of selecting 10 out of 17 regions? (3 pts) 4. Expand the (2𝑥 − 𝑦)3 using binomial coefficient. (14 pts) 5. Find for the coefficient of the following: (3 pts each) a. x4y3 after the expansion of (x – 2y )7 b. x6y6 after the expansion of (2x + y )12 6. Find the 9th term of the expression (2x + y )12 (3 pts)**

The **Answer to the Question**

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**Here's the Solution to this Question**

1.a) 5*5*5=125

1.b) 5*4*3=60

2) 8*7*6*5=1 680

3) $C^{10}_{17}=19448$

4) binomial coefficients:1 3 3 1

$(2x-y)^3=(2x)^3*1+(2x)^2*(-y)*3+(2x)*(-y)^2*3+(-y)^3*1=\\ =8x^3-12x^2y+6xy^2-y^3$

5.a) binomial coefficients: 1 7 21 35 35 21 7 1

$35*(x)^4*(-2y)^3=280 x^4y^3$

5.b) binomial coefficients: 1 12 66 220 495 792 924 792 495 220 66 12 1

$924 *(2x)^6(y)^6=59136 x^6y^6$

6) binomial coefficients: 1 12 66 220 495 792 924 792 495 220 66 12 1

$495 *(2x)^{12-9+1}(y)^{9-1}=495 *(2x)^4(y)^8=7920x^4y^8$