You have the Scrabble hand SYMPTOM To form wont you need to

You have the Scrabble hand \"SYMPTOM.\" To form wont>>, you need to pick some of these letters and arrange them in an order. (Each letter tile can be used only once, so you can use two M\'s in your word but only one O.) How many 7-letter \"words\" (strings - not necessarily English words} can be made using these letters? How many 4-letter \"words\" can you make without using any M\'s? How many 4-letter sub-hands can you make with two M\'s in them. How many 4-letter \"words\" can you make with two M\'s In them? How many 4-letter \"words\" can you make with one M? How many total 4-letter \"words\" can you make?

Solution

Word: SYMPTOM

(a) 7-letter word: To fill 1st place we have 7 options (all letter)

To fill 2nd place we have 6 options (all letter expect the one filled in 1st place)

...............

To fill last place we have 1 option

So total number of words: 7*6*...*2*1=7!

(b) Word without M: SYPTO

To fill 1st place we have 5 options (all letter)

To fill 2nd place we have 4 options

To fill 3rd place we have 3 options

To fill 4th place we have 2 options

Total Number of words:= 5*4*3*2=120

(c) 4 letter sub-hand with two M:

To fill 1st place we have 6 options

To fill 2nd place we have 5 options

To fill 3rd place we have 4 options

To fill 4th place we have 3 options

Total= 6*5*4*3=360

(d) 4 letter word with two M

To fill 1st place we have 6 options

To fill 2nd place we have 5 options

To fill 3rd place we have 4 options

To fill 4th place we have 3 options

Total= 6*5*4*3=360