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Permutations and Combinations

Multiple Choice Questions

11.

The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together, is

$\frac{7!}{2! 2!}$
$\frac{7!}{2!}$
$\frac{6!}{2!}$
$5! \times 2!$

12.
If $\frac{1}{{}^{5}C_{r}} + \frac{1}{{}^{6}C_{r}} = \frac{1}{{}^{4}C_{r}}$ , then the value of r is

4

2

5

3

Given, $\frac{1}{{}^{5}C_{r}} + \frac{1}{{}^{6}C_{r}} = \frac{1}{{}^{4}C_{r}}$

$\Rightarrow \frac{r! (5 - r)!}{5!} + \frac{r! (6 - r)!}{6!} = \frac{r! (4 - r)!}{4!}$

$\Rightarrow \frac{r! (5 - r)!}{5} + \frac{r! (6 - r)!}{30} = (4 - r)!$

$\Rightarrow \frac{(5 - r)}{5} + \frac{(6 - r) (5 - r)}{30} = 1$

$\Rightarrow 30 - 6 r + 30 - 1 r + r^{2} = 30$

$\Rightarrow r^{2} - 17 r + 30 = 0$

$\Rightarrow r^{2} - 15 r - 2 r + 30 = 0$

$\Rightarrow r (r - 15) - 2 (r - 15)$

$\Rightarrow (r - 2) (r - 15)$

$\Rightarrow r = 15, 2$

Hence, the value of r is 2.

13.

If A = $\{5^{n} - 4 n - 1 : n \in N\}$ and $B = \{16 (n - 1 : n \in N)\}$ , then

A = B
$A \cap B = φ$
$A \subseteq B$
$B \subseteq A$

14.

The letters of the word COCHIN are permuted and all permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

15.

The letters of the word 'COCHIN' are permuted and all the permutations are arranged in alphabetical order as in English dictionary. The number of words that appear before the word 'COCHIN', is

16.

The number of digits in 20³⁰¹ $(given, \log_{10} (2) = 0.3010)$ is

17.

The number of solutions of the equation x + y + z = 10 where x, y and z are positive integers

18.

Let n be a positive even integer. If the ratio of the largest coefficient and the 2nd largest coefficient in the expansion of (1 + x)ⁿ is 11 : 10. Then, the number of terms in the expansion of (1 + x)ⁿ is

19.

There are 100 students in a class. In an examination, 50 of them failed in Mathematics, 45 failed in Physics, 40 failed in Biology and 32 failed in exactly two of the three subjects. Only one student passed in all the subjects. Then, the number of students failing in all the three subjects