If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is ( 6 ! ) k then k is equal to

Step-by-step solution

The word 'MATHEMATICS' consists of 11 letters including two M ' s , two T ' s and two A ' s , Now, first we will arrange letters _ M _ A _ T _ H _ E _ M _ A _ T _ I _ . Here, we have 10 gaps to fill C   &   S . The number of words with or without meaning can be formed from the word MATHEMATICS when C ,   S does not come together is = 9 ! 2 ! · 2 ! · 2 ! × C 2 10 × 2 ! = 9 × 8 × 7 × 6 ! 2 ! · 2 ! × 10 × 9 2 = 5670 Hence this is the correct