NEET Test Series from KOTA - 10 Papers In MS WORD
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Permutation and Combination
119197
In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes?
1 1024
2 625
3 120
4 600
Explanation:
A Here, First prize may be given to any one of the 4 boys, hence first prize can be distributed in 4 ways. Similarly every one of second, third, fourth and fifth prizes can also be given in 4 ways. \(\therefore\) The number of ways of their distribution, \(=4 \times 4 \times 4 \times 4 \times 4\) \(=4^5\) \(=1024\)
BITSAT-2012
Permutation and Combination
119199
900 distinct n-digit positive numbers are to be formed using only the digit 2,5 and 7 . The smallest value of \(n\) for which this is possible, is
1 9
2 8
3 7
4 6
Explanation:
C : Given digits \(=2,5,7\) The total number of ways to make number of \(n\)-digit number \(=3^{\mathrm{n}}\) According to question \(3^{\mathrm{n}} \geq 900\) \(\frac{3^{\mathrm{n}}}{3^2} \geq 100\) \(3^{\mathrm{n}-2} \geq 100\) Suppose that \(\mathrm{n}=7 \quad 3^4=81\) and \(3^5=243\) Smallest value of \(\mathrm{n}\) for which 900 distinct number can be formed \(\mathrm{n}>7\) Smallest value of \(\mathrm{n}=7\)
JCECE-2016
Permutation and Combination
119200
The number of six-digit numbers that can be formed from the digits \(1,2,3,4,5,6\) and 7 , so that digits do not repeat and the terminal digits are even is
1 144
2 72
3 288
4 720
Explanation:
D : Terminal digits are the first and last digits. Given, the terminal digit are even. Since, digit in a number are not repeated Therefore, first place can be filled in 3 ways and last place can filled in 2 ways. Remaining places can be filled in \({ }^5 \mathrm{P}_4=120\) Hence, the number of six digit number such that the terminal digit are even \(=3 \times 120 \times 2=720\)
BCECE-2013
Permutation and Combination
119201
What is the number of ways in which sum of upper faces of four distinct dice can be six?
1 10
2 7
3 6
4 4
Explanation:
A According to question, \(\begin{array}{llll}\mathrm{x}_1 \mathrm{x}_2 \mathrm{x}_3 \mathrm{x}_4 \\ =6\end{array}\) \(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=6\) \(1 \leq \mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4 \leq 6\) Here, \(\mathrm{n}=6, \mathrm{r}=4\) Thus, the number of ways \(={ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}={ }^{6-1} \mathrm{C}_{4-1}={ }^5 \mathrm{C}_3=\frac{5 \times 4 \times 3 !}{3 ! \times 2 !}=10\)
119197
In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes?
1 1024
2 625
3 120
4 600
Explanation:
A Here, First prize may be given to any one of the 4 boys, hence first prize can be distributed in 4 ways. Similarly every one of second, third, fourth and fifth prizes can also be given in 4 ways. \(\therefore\) The number of ways of their distribution, \(=4 \times 4 \times 4 \times 4 \times 4\) \(=4^5\) \(=1024\)
BITSAT-2012
Permutation and Combination
119199
900 distinct n-digit positive numbers are to be formed using only the digit 2,5 and 7 . The smallest value of \(n\) for which this is possible, is
1 9
2 8
3 7
4 6
Explanation:
C : Given digits \(=2,5,7\) The total number of ways to make number of \(n\)-digit number \(=3^{\mathrm{n}}\) According to question \(3^{\mathrm{n}} \geq 900\) \(\frac{3^{\mathrm{n}}}{3^2} \geq 100\) \(3^{\mathrm{n}-2} \geq 100\) Suppose that \(\mathrm{n}=7 \quad 3^4=81\) and \(3^5=243\) Smallest value of \(\mathrm{n}\) for which 900 distinct number can be formed \(\mathrm{n}>7\) Smallest value of \(\mathrm{n}=7\)
JCECE-2016
Permutation and Combination
119200
The number of six-digit numbers that can be formed from the digits \(1,2,3,4,5,6\) and 7 , so that digits do not repeat and the terminal digits are even is
1 144
2 72
3 288
4 720
Explanation:
D : Terminal digits are the first and last digits. Given, the terminal digit are even. Since, digit in a number are not repeated Therefore, first place can be filled in 3 ways and last place can filled in 2 ways. Remaining places can be filled in \({ }^5 \mathrm{P}_4=120\) Hence, the number of six digit number such that the terminal digit are even \(=3 \times 120 \times 2=720\)
BCECE-2013
Permutation and Combination
119201
What is the number of ways in which sum of upper faces of four distinct dice can be six?
1 10
2 7
3 6
4 4
Explanation:
A According to question, \(\begin{array}{llll}\mathrm{x}_1 \mathrm{x}_2 \mathrm{x}_3 \mathrm{x}_4 \\ =6\end{array}\) \(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=6\) \(1 \leq \mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4 \leq 6\) Here, \(\mathrm{n}=6, \mathrm{r}=4\) Thus, the number of ways \(={ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}={ }^{6-1} \mathrm{C}_{4-1}={ }^5 \mathrm{C}_3=\frac{5 \times 4 \times 3 !}{3 ! \times 2 !}=10\)
119197
In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes?
1 1024
2 625
3 120
4 600
Explanation:
A Here, First prize may be given to any one of the 4 boys, hence first prize can be distributed in 4 ways. Similarly every one of second, third, fourth and fifth prizes can also be given in 4 ways. \(\therefore\) The number of ways of their distribution, \(=4 \times 4 \times 4 \times 4 \times 4\) \(=4^5\) \(=1024\)
BITSAT-2012
Permutation and Combination
119199
900 distinct n-digit positive numbers are to be formed using only the digit 2,5 and 7 . The smallest value of \(n\) for which this is possible, is
1 9
2 8
3 7
4 6
Explanation:
C : Given digits \(=2,5,7\) The total number of ways to make number of \(n\)-digit number \(=3^{\mathrm{n}}\) According to question \(3^{\mathrm{n}} \geq 900\) \(\frac{3^{\mathrm{n}}}{3^2} \geq 100\) \(3^{\mathrm{n}-2} \geq 100\) Suppose that \(\mathrm{n}=7 \quad 3^4=81\) and \(3^5=243\) Smallest value of \(\mathrm{n}\) for which 900 distinct number can be formed \(\mathrm{n}>7\) Smallest value of \(\mathrm{n}=7\)
JCECE-2016
Permutation and Combination
119200
The number of six-digit numbers that can be formed from the digits \(1,2,3,4,5,6\) and 7 , so that digits do not repeat and the terminal digits are even is
1 144
2 72
3 288
4 720
Explanation:
D : Terminal digits are the first and last digits. Given, the terminal digit are even. Since, digit in a number are not repeated Therefore, first place can be filled in 3 ways and last place can filled in 2 ways. Remaining places can be filled in \({ }^5 \mathrm{P}_4=120\) Hence, the number of six digit number such that the terminal digit are even \(=3 \times 120 \times 2=720\)
BCECE-2013
Permutation and Combination
119201
What is the number of ways in which sum of upper faces of four distinct dice can be six?
1 10
2 7
3 6
4 4
Explanation:
A According to question, \(\begin{array}{llll}\mathrm{x}_1 \mathrm{x}_2 \mathrm{x}_3 \mathrm{x}_4 \\ =6\end{array}\) \(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=6\) \(1 \leq \mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4 \leq 6\) Here, \(\mathrm{n}=6, \mathrm{r}=4\) Thus, the number of ways \(={ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}={ }^{6-1} \mathrm{C}_{4-1}={ }^5 \mathrm{C}_3=\frac{5 \times 4 \times 3 !}{3 ! \times 2 !}=10\)
119197
In how many ways can 5 prizes be distributed among 4 boys when every boy can take one or more prizes?
1 1024
2 625
3 120
4 600
Explanation:
A Here, First prize may be given to any one of the 4 boys, hence first prize can be distributed in 4 ways. Similarly every one of second, third, fourth and fifth prizes can also be given in 4 ways. \(\therefore\) The number of ways of their distribution, \(=4 \times 4 \times 4 \times 4 \times 4\) \(=4^5\) \(=1024\)
BITSAT-2012
Permutation and Combination
119199
900 distinct n-digit positive numbers are to be formed using only the digit 2,5 and 7 . The smallest value of \(n\) for which this is possible, is
1 9
2 8
3 7
4 6
Explanation:
C : Given digits \(=2,5,7\) The total number of ways to make number of \(n\)-digit number \(=3^{\mathrm{n}}\) According to question \(3^{\mathrm{n}} \geq 900\) \(\frac{3^{\mathrm{n}}}{3^2} \geq 100\) \(3^{\mathrm{n}-2} \geq 100\) Suppose that \(\mathrm{n}=7 \quad 3^4=81\) and \(3^5=243\) Smallest value of \(\mathrm{n}\) for which 900 distinct number can be formed \(\mathrm{n}>7\) Smallest value of \(\mathrm{n}=7\)
JCECE-2016
Permutation and Combination
119200
The number of six-digit numbers that can be formed from the digits \(1,2,3,4,5,6\) and 7 , so that digits do not repeat and the terminal digits are even is
1 144
2 72
3 288
4 720
Explanation:
D : Terminal digits are the first and last digits. Given, the terminal digit are even. Since, digit in a number are not repeated Therefore, first place can be filled in 3 ways and last place can filled in 2 ways. Remaining places can be filled in \({ }^5 \mathrm{P}_4=120\) Hence, the number of six digit number such that the terminal digit are even \(=3 \times 120 \times 2=720\)
BCECE-2013
Permutation and Combination
119201
What is the number of ways in which sum of upper faces of four distinct dice can be six?
1 10
2 7
3 6
4 4
Explanation:
A According to question, \(\begin{array}{llll}\mathrm{x}_1 \mathrm{x}_2 \mathrm{x}_3 \mathrm{x}_4 \\ =6\end{array}\) \(\mathrm{x}_1+\mathrm{x}_2+\mathrm{x}_3+\mathrm{x}_4=6\) \(1 \leq \mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4 \leq 6\) Here, \(\mathrm{n}=6, \mathrm{r}=4\) Thus, the number of ways \(={ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}={ }^{6-1} \mathrm{C}_{4-1}={ }^5 \mathrm{C}_3=\frac{5 \times 4 \times 3 !}{3 ! \times 2 !}=10\)
