The sum of the digits at the 10^th place of all numbers formed with the help of 2, 4, 5, 7 taken all at a time is

The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

The product of r consecutive positive integers is divisible by

The number of five digit telephone numbers having at least one of their digits repeated is

The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is

A person went to a restaurant for dinner. In the menu card, the person saw 10 Indian and 7 Chinese food items. In how many ways the person can select either an Indian or a Chinese food?

Given four flags of different colours, how many different signals can be generated if each signal requires the use of three flags, one below the other?

Find the value of \(\frac { 12! }{ 9!\times 3! } \)

If ¹⁰P_r-1 = 2 \(\times\) ⁶P_r, find r.

Prove that ³⁵C₅ + \(\sum _{ r=0 }^{ 4\quad (39-r) }{ C_{ 4 }=^{ 40 }{ C }_{ 5 } } \)

A polygon has 90 diagonals. Find the number of its sides?

Count the numbers between 999 and 10000 subject to the condition that there are
(i) no restriction.
(ii) no digit is repeated.
(iii) at least one of the digits is repeated.

Using the mathematical induction, show that for any natural number n
\({1\over 2.5}+{1\over 5.8}+{1\over 8.11}+...+{1\over (3n-1)(3n+2)}={n\over 6n+4}\)