Permutations and Combinations Test 43

Directions For Questions

Let $$p$$ be a prime number & $$n$$ be a positive integer, then exponent of prime $$p$$ in $$n!$$ is denoted by $$\displaystyle E_{p}(n!)$$ & is given by $$\displaystyle E_{p}(n!)=\left[\frac{n}{p}\right]+\left[\frac{n}{p^{2}}\right]+\left[\frac{n}{p^{3}}\right]+.....+\left[\frac{n}{p^{x}}\right]$$ where $$x$$ is the largest positive integer such that $$\displaystyle p^{x}\leq n<p^{x+1}$$ and $$\displaystyle [\cdot ] $$ denotes the greatest integer
Again every natural number $$N$$ can be expressed as the product of its prime factors given by $$\displaystyle N=P_{1}^{k_{2}}P_{2}^{k_{2}}....P_{r}^{k_{r}}$$ where $$\displaystyle P_{1},P_{2},P_{3},.......P_{r}$$ are prime numbers & $$\displaystyle k_{1}$$ are whole numbers.

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Permutations an...

Question 1

$$\displaystyle 105^2$$

$$\displaystyle 210 \: \times \: 243$$

$$\displaystyle 105 \: \times \: 243$$

none of these

Question 2

$$\displaystyle ^{51}C_{20}$$

$$\displaystyle 2 . ^{50}C_{20}$$

$$2 . ^{45}C_{15}$$

none of these

Question 3

$$\displaystyle 3\times 2^4$$

$$\displaystyle 2 \times 4^4$$

$$\displaystyle 3 \times 4^4$$

none of these

Question 4

$$\displaystyle n\cdot 2^{n-1}-1$$

$$\displaystyle n\cdot 2^{n-1}+(n+1)!$$

$$\displaystyle n\cdot 2^{n-1}+(n+1)!-1$$

$$\displaystyle n^{2}+n+5$$

Question 5

$$225$$

$$325$$

$$215$$

$$315$$

Question 6

$$ ^{47}C_{5} $$

$$ ^{52}C_{5} $$

$$ ^{52}C_{4} $$

$$ ^{52}C_{3} $$

Question 7

0

1

2

3

Question 8

$$\dfrac{2^{20}}{21}$$

$$\dfrac{2^{19}}{21}$$

$$\dfrac{2^{20}}{3\times 77}$$

$$\dfrac{2^{19}}{33\times 7}$$

Question 9

$$1$$

$$2$$

$$4$$

None of these

Question 10

$$ ^{n+3}C_{r-1} $$

$$ ^{n+2}C_{r+1} $$

$$ ^{n+4}C_{r+1} $$

$$ ^{n+1}C_{r-1} $$

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