If ⁴⁰C_r+2 = ⁴⁰C_r-2 then r is equal to ______.

If ^mC₂ = ⁿC₁ then ______.

If ⁿC_r+ⁿC_r+1=ⁿ⁺¹C_x then x is equal to ______.

If C₀ + C₁ + C₂ +...+C_n = 256 then ²ⁿC₂ is equal to ______.

If ⁿ⁺¹C₃ = 2.ⁿC₂ then n is equal to ______.

How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?

In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

How many three digit odd numbers can be formed by using the digits 1,2,3,4,5 and 6 if (i) the repetition of digits is not allowed. (ii) the repetition of digits is allowed.

In how many ways can 5 letters be posted in 4 letter boxes?

Compute \(\frac { 8! }{ 6!\times 2! } \)

If \(\frac { 1 }{ 6! } +\frac { 1 }{ 7! } =\frac { x }{ 8! } \), find x

In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together ?

A gentleman has 5 friends to invite. In how many ways can he send invitation cards to them, if he has three servants to carry the cards?

How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated ?

If ⁿP_r=ⁿP_r+1 and ⁿC_r = ⁿC_r-1, find the values of n and r.

Therre are 2 candidates for Classical language, 5 for Mathematics and 4 for Natural Science scholarship.
(i) In how many ways can these scholarship be awarded?
(ii) In how many ways, one of these scholarship be awarded?

Find the number of different 8 letters of the word 'DAUGHTER' so that 
(i) all vowels occur together.

If \(\frac { (2n)! }{ 3!(2n-3)! } \) and \(\frac { n! }{ 2!(n-2)! } \) are in the ration 44:3, find n

In how many ways can the letters of the word PERMUTATIONS be arranged if the
(i) words start with P and end with S
(ii) vowels are all together
(iii) there are always 4 letters between P and S?