# First Year Math Permutation, Combination and Probability 4. How many ( a ) diagonals and (b) triangles can be formed by joining the vertices of the polygon having:i) 5 sides

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##### 4. How many ( a ) diagonals and (b) triangles can be formed by joining the vertices of the polygon having:i) 5 sides

Q.48 If { }^{n} C_{3}={ }^{n} C_{12} then the value of n equal to :(a) 4(b) 8(c) 20(d) 12

8. Find the numbers greater than 23000 that can be formed from the digits 12356 without repeating any digit.HINT: The first two digits on L.H.S. will be 23 etc.

2. Write each of the following in the factorial form:iii) 20.19 .18 .17

Q.26 { }^{n} C_{r} is equal to :(a) \frac{n !}{(n-r) ! r !} (b) \frac{n !}{(n-r) !} (c) \frac{n !}{r !} (d) \frac{(n-r) ! r !}{n !}

6. How many words can be formed from the letters of the following words using all letters when no letter is to be repeated:iii) FASTING?

2. Write each of the following in the factorial form: i) 6.5 .4 .

11. In how many ways can 4 keys be arranged on a circular key ring?

12. In how many ways can 8 books including 2 on English be arranged on a shelf in such a way that the English books are never together?

1. Evaluate each of the following:iv) \frac{10 !}{7 !}

2. Write each of the following in the factorial form:viii) (n+2)(n+1)(n)

Q.44 { }^{8} C_{3}= (a) { }^{8} P_{3} (b) { }^{8} C_{5} (c) { }^{3} C_{B} (d) { }^{5} C_{8} Rawalpindi Board 2011

1. Evaluate each of the following:vi) \frac{6 !}{3 ! 3 !}

3. How many arrangements of the letters of the word ATTACKED can be made if each arrangement begins with \mathrm{C} and ends with \mathrm{K} ?

13. Find the number of arrangements of 3 books on English and 5 books on Urdu for placing them on a shelf such that the books on the same subject are together.

4. How many numbers greater than 1000000 can be formed from the digits 02223 44 ?

Q.34 If a die is rolled then n(S) equals :(a) 36 \begin{array}{ll}\text { (b) } 6 & \text { (c) } 1\end{array} (d) 9

Q.40 The sample space for tossing a coin once is :(a) \{H\} (b) \{T\} (c) \{H H\} (d) \{H T\}

Example 1: A die is rolled. What is the probability that the dots on the top are greater than 4 ?

2. Write each of the following in the factorial form:x) n(n-1)(n-2) \ldots .(n-r+1)

Q.47 If { }^{n} C_{12}={ }^{n} C_{\theta} then the value of n is :(a) 12 \begin{array}{ll}\text { (b) } 14 & \text { (c) } 16\end{array} (d) 18Rawalpindi Board 2013

Q.37 Range of the probability of an event E is(a) (01) \begin{array}{ll}\text { (b) }[01] & \text { (c) }[01)\end{array} (d) (01]

Example 2: Two dice are thrown. E_{1} is the event that the sum of their dots is an odd number and E_{2} is the event that 1 is the dot on the top of the first die. Show that P\left(E_{1} \cap E_{2}\right)=P\left(E_{1}\right) \cdot P\left(E_{2}\right)

12. How many necklaces can be made from 6 beads of different colours?

3. Find the values of n and r whenii) { }^{n-1} C_{n 1}:{ }^{n} C_{r}:{ }^{n+1} C_{r+1}=3: 6: 11

Q.32 With usual notation { }^{6} C_{2} equals(a) 30(b) 20(c) 12(d) 15

Q.9 Factorial form of n(n-1)(n-2)= (a) \frac{n !}{(n-1) !} (b) \frac{n !}{(n-2) !} (c) \frac{n !}{(n-3) !} (d) \frac{n !}{(n+3) !}