application of multinomial in permutation with repetition

Application of multinomial in permutation with repetition:-

⦿ Number of selection/combination of r things out of n things of which p are alike of one kind & q are alike of second kind and remaining (n-p-q) things are all different is given by 
⟹ coefficient of xʳ in (1+x+x²+……..+xᵖ)(1+x+x²+……..+xᑫ){(1+x)(1+x)……… till (n-p-q) times}

⦿ The number of permutation/arrangement of r things out of n things of which p are alike of one kind & q are alike of second kind and so on……….

Example:- Let there are
                                p, q ━ 5
                                r, s ━ 4
                                    t ━ 3
                          u, v, w ━ 1
then find the number of permutation/arrangement of 5 letter words selected from given letters
Solution:- 

Using Multinomial theorem:-

Method(2):       Listing all cases: 

Example:- In how many ways 4 letters of the word ‘PARALLEL’ can be selected & how many 4 letter words can be formed?
Solution:-

L ━ 3        A ━ 2        P ━ 1        R ━ 1        E ━ 1
∴ Required number of ways sof selection:-

Example:- Find the number of 5-letter words that can be formed from the letters given below:-

p, q, r ━ 6
     s, t ━ 5
     u, v ━ 4
     w, y ━ 3
Solution:- 

Method(1):

Required number of 5-letter words:-