Permutations Combinations 2-1

1How many arrangements can be made of the letters of the word “ASSASSINATION”? In how many of them are the vowels always together?
A$\dfrac{{13!}}{{{{\left( {4!} \right)}^2}}},\;\dfrac{{8! \times 6!}}{{{{\left( {4!} \right)}^2}}}$
B$\dfrac{{13!}}{{6! \times 7!}},\;\dfrac{{8! \times 6!}}{{{{\left( {4!} \right)}^2}}}$
C$\dfrac{{13!}}{{6! \times 7!}},\;\dfrac{{8! \times 6!}}{{6! \times 7!}}$
D$\dfrac{{13!}}{{{{\left( {4!} \right)}^2}}},\;\dfrac{{8! \times 6!}}{{6! \times 7!}}$


2In how many ways can the letters of the word ARRANGE be arranged so that two R’s are never together
A900
B360
C120
D1260


3In how many ways can the letters of the word ARRANGE be arranged so that The Two A’s are together but not two R’s
A900
B360
C240
D1260


4In how many ways can the letters of the word ARRANGE be arranged so that neither Two A’s nor two R's are together
A900
B360
C120
D660


5Ten different alphabets are given. Words containing five alphabets are to be formed from them. Find the number of words which have exactly one alphabet repeats.
A${}^{10}{P_5}$
B${\rm{10}}^{\rm{5}} $
C${10^5}{ - ^{10}}{P_5}$
D$58060$


6How many 4 letter words may be formed by using the letters of the word 'ASSASSINATION'
A916
B917
C360
D480