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The number of ways 5 letter be put in 3 letter boxes A,B,C. If letter box A must contain at least 2 letters.

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 5 letters be put into 3 letter boxes A,B,C...

// means 5 different letters in 3 different boxes..

condition : letter box A contains at least 2 letters...

method 1:

required ways where letter box A has at least 2 letters= box A has 2 letters + box A has #letters + box A has 4 letters + box A has 5 letters

                         =5C2*2^3 + 5C3*2^2+5C4*2^1 +5C5

                        =80+40+10+1 =131 answer...

shortcut method:

required ways where letter box A has atleast 2 letters=total ways - at most one letter in box A

= total number of ways in which each letter can be assigned to any letter box - when letter box A has 1 letter - letter box A has no letter.

=> 3^5 - 5C1*2^4 - 5C0*2^5

=243- 5*16 - 32

=243-112 = 131 answer.... 

 

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The answer depends on weather the letters are same or different. If the letters are same answer is 10, if different answer is 131.
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correct answer is 10.

This problem belongs to combinations with repetitions. Try the examples from rosen's book

 

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