${}^{37}\mathrm{C}_4 + \sum _{\mathrm{r} = 1}^5{}^{\left(42 - \mathrm{r}\right)}\mathrm{C}_{\mathrm{r}} = ?$

• ${}^{41}\mathrm{C}_4$

• ${}^{39}\mathrm{C}_4$

• ${}^{38}\mathrm{C}_4$

• ${}^{42}\mathrm{C}_4$

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ...

For a positive integer $\mathrm{n}, {\left(1 + \frac{1}{\mathrm{x}}\right)}^{\mathrm{n}}$ is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12 , then n is equal to _____

A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is ......

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coffee and tea, then x cannot be :

• 36

• 63

• 38

• 54

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are a like, is _________.

There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least one question from each section. Then the number of ways, in which the candidate can choose the questions, is :

• 3000

• 2250

• 2255

• 1500

$\mathrm{If} \left\{\mathrm{p}\right\} \mathrm{denotes} \mathrm{the} \mathrm{fractional} \mathrm{part} \mathrm{pf} \mathrm{the} \mathrm{number} \mathrm{p}, \mathrm{then} \left\{\frac{3^{200}}{8}\right\} = ?$

• $\frac{1}{8}$

• $\frac{3}{8}$

• $\frac{7}{8}$

• $\frac{5}{8}$

The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is...

Consider the data on x taking the values 0, 2, 4, 8, ...., 2ⁿ with frequencies ⁿC₀, ⁿC₁, ⁿC₂,..., ⁿC_n respectively. If the mean of this data is $\frac{728}{2^{\mathrm{n}}}$, then n is equal to....