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First Year Math Permutation, Combination and Probability Q.15 { }^{6} P_{3}= (a) 36(b) 360(c) 6(d) 120


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Q.15 { }^{6} P_{3}= (a) 36(b) 360(c) 6(d) 120

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

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.
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.

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

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 !}
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 !}

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?
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?

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

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

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?
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?

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

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

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

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

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} ?
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}  ?

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} ?

Q.15 { }^{6} P_{3}= (a) 36(b) 360(c) 6(d) 120
Q.15  { }^{6} P_{3}= (a) 36(b) 360(c) 6(d) 120
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Q.15 { }^{6} P_{3}= (a) 36(b) 360(c) 6(d) 120

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.
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.

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

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.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.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

1. Evaluate the following:iv) { }^{10} P_{7}
1. Evaluate the following:iv)  { }^{10} P_{7}

1. Evaluate the following:iv) { }^{10} P_{7}

Q.40 The sample space for tossing a coin once is :(a) \{H\} (b) \{T\} (c) \{H H\} (d) \{H T\}
Q.40 The sample space for tossing a coin once is :(a)  \{H\} (b)  \{T\} (c)  \{H H\} (d)  \{H T\}
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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 ?
Example 1: A die is rolled. What is the probability that the dots on the top are greater than  4 ?
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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)
2. Write each of the following in the factorial form:x)   n(n-1)(n-2) \ldots .(n-r+1)
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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.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
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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]
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]
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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)
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)
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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?
12. How many necklaces can be made from 6 beads of different colours?
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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
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
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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

4. How many ( a ) diagonals and (b) triangles can be formed by joining the vertices of the polygon having:i) 5 sides
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.32 With usual notation { }^{6} C_{2} equals(a) 30(b) 20(c) 12(d) 15
Q.32 With usual notation  { }^{6} C_{2}  equals(a) 30(b) 20(c) 12(d) 15
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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) !}
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) !}
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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) !}

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