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22-52 26 ' 16 1600 5 . gb za = = - . 63 E I 2 0 E . - - a , , , 23 lo total I b e , , ways = f # options for 5) 2) ( # options for of . 3=1217 t I ) ( 2 I ) ( 6 7 = t = I 2 19 factors 2 : = - proper

=/3 I EG 3-63-6 . . . .

S M C ∣ S ∪ M ∪ C ∣ = ∣ S ∣ + ∣ M ∣ + ∣ C ∣ = 3 + 4 + 2 = 9 n , n , ... n k 1 2 k n + n + ... + n 1 2 k -

An BI At tf BI =/ IA UBI I - 25 TO 40 x 50 = 40 + 25 − x ⇒ x = 15

: 1333J - I 3 333 of 2 3 . - - - 3 - . Mutt 3 : , , 5 200 51,5-2 - of 5 malt - : - - - , 66J :/ 15-1,15-2 15%6 15 malt of . - - : , 333+200-66=14677 Total :

! 5 -3-21 I = 5- ⇒ diff 473£ Go -3 I 5- =

n ! = n ⋅ ( n − 1) ⋅ ... ⋅ 2 ⋅ 1 k n n ! P ( n , k ) = ( n − k )! i÷n 5⋅4⋅3⋅2⋅1 5! 5 ⋅ 4 ⋅ 3 = = 2⋅1 2! . - n ! n n ! P ( n , n ) = = n ! 0!

Pilar RADI D 3! = 6 Dz AD , A Dz D , ! R of D : how 2 permutations many , RADAR . . - . . DADD : 3 !

I M : ! 11 4 : - ssss 'D 4714114 ( 1111 4 : " repeat Tnepeasted Ps 2 repeated PP :

! BB 2 A Y DOG of characters the do times How many other next each to , appear order : any :O . : :* : : : I

but gray ;;t rotated - ÷ , . 5 ! , 4 ! = I

5! 2! 3! = 6 5! 3! 2! 3! 5! 5 ⋅ 4 ⋅ 3! = = 10 2!3! 2 ⋅ 1 ⋅ 3!

k n n ! ( k n ) = ( n − k )! k ! n k n ) n ) = n − k ( k ( n − k k { A , B , C , D , E } C 5 ) C II - - - C : ) ( it -

O O O 00 - - 8 ) ( 3 9 ) ( 5 8 )( 5 9 ) ( 3

8 − k k ways (at least 4 b) = ways (4 b) + ways (5 b) + ways (6 b) + ways (7 b) + ways (8 b) 8 )( 4 9 ) 8 )( 3 9 ) 8 )( 2 9 ) 8 )( 1 9 ) 8 )( 0 9 ) ( 4 ( 5 ( 6 ( 7 ( 8 = + + + +

Find I I } ) . ( 42 ) - . 43 ( 13 . I # ¥ ÷÷ #÷¥÷ I . .

EE ' EE 4¥ . . . 9. 4-4.4-4.4=19-457 g. I Y I - 44k¥17

= ( ntl ) It 2T th - . . . I tht th S = - - - proof - Ht nth S= I t - - . - combinatorial Show LHS and ( I ) KHS count ABCDE - same quantity AB 4 AI.AD.AE , BE 3 BD BE , , 2 CE CD , DE I