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Tom Riddle must choose 4 classes for his final semester at school. He must take at least 1 science class and at least 1 art class. In his school offers 4 (distinct) science classes, 3 (distinct) art classes and three other (distinct) classes, hoe many different choices for classes does we have? ________

  1.   195
  2.   175
  3.   210
  4.   160
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10C​​​​​​(total ways) - 7C (0 class from Arts)- 6C4  (0 Class from Science)

= 160

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Ok,

Let's first make all combination and then solve each combination and add all to get the final result

 

All the cases which involves atleast one science and atlease one art , and total subject choosen have to be four :

Science              Art                 Other                             Total

C( 4 , 1 )    *     C(  3 , 1 )      *      C( 3 , 2 )                   36   

C( 4 , 1 )    *     C(  3 , 2 )     *      C( 3 , 1 )                    36

C( 4 , 1 )    *     C(  3 , 3 )     *      C( 3 , 0 )                      4

C( 4 , 2 )    *     C(  3 , 1 )     *      C( 3 , 1 )                    54

C( 4 , 2 )    *     C(  3 , 2 )     *      C( 3 ,  0 )                   18

C( 4 , 3 )    *     C(  3 , 1 )     *      C( 3 ,  0 )                   12

Total ways    :   36 + 36 + 4 + 54 + 18 + 12 :  160 ways

So,  anwer will be (D) option

We don't multiply the answer to 4!  because classes are same,  (Science1 , Art1 , Other1 , Other2 )  and

(Science1 , Other1 , Art1 , Other2 ) both are same, hence not counted as different.

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