Permutations an...

If $$\dfrac { ^{ n }{ P }_{ r-1 } }{ a } =\dfrac { ^{ n }{ P }_{ r } }{ b } =\dfrac { ^{ n }{ P }_{ r+1 } }{ c } $$, then which of the following holds good:

If $$^nC_r : ^nC_{r + 1} : ^nC_{r + 2} = 3 : 4: 5$$, then the value of $$2n + 3r$$ is

The number of permutations of the letters of the word $$HONOLULU$$ taken $$4$$ at a time is

The total number of different combinations of one
or more letters which can be made from the letter of the word MISSISSIPPI is,

Find the remainder when $$105!$$ is divided by $$214.$$ 

We wish to select $$6$$ persons from $$8$$, but if the person A is chosen, then B must be chosen. In how many ways can selections be made?

Number of arrangements of the letter $$HOLLYWOOD$$ in which all $$Os$$ are separated.

$$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\dfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\dfrac { { C }_{ 3 } }{ { C }_{ 2 } } +....+15\dfrac { { C }_{ 15 } }{ { C }_{ 14 } }$$ is equal to

How many committee of five persons with a chairperson can be selected from $$12$$ persons.

The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet ?