Question about 50 lectures book 2

[lele_2013]

Could you please post solutions to these problems?

Problems from 50 Lectures Book 2(page 102)

Problem 16. How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5?
Ans. 4970

Problem 38. Twelve students are to be divided among Mr. Mirus’s and Ms. Batty’s classes.  No teacher is to have
more than 8 students. How many different groups of students could be in  Mr. Mirus’s class? (2001 Mathcounts Handbook).
Ans.3498

[yongcheng3315]

Could you please post solutions to these problems?

Problems from 50 Lectures Book 2(page 102)



Problem 16. How many four-digit numbers greater than 2999 can be formed such that the product of the middle two digits exceeds 5?
Ans. 4970


Solution:
The thousands digits can be 3, 4, 5, 6, 7, 8, 9. 7 ways.
The last digit can be 0 to 9. 10 ways.

The middle two digits:

hundreds digt is 1 and tens digit can be 6, 7, 8,9: 4 ways.
hundreds digit is 2 and tens digit can be 3, 4, ...9: 7 ways.
hundreds digit is 3 and tens digit can be 2, 3,...9: 8 ways.
................................
Simialrly, we get 4 + 7 +8 + 8 +8+9+9+9+9 = 71 ways.

The answer is 7 * 71*10 = 4970.



Problem 38. Twelve students are to be divided among Mr. Mirus’s and Ms. Batty’s classes.  No teacher is to have
more than 8 students. How many different groups of students could be in  Mr. Mirus’s class? (2001 Mathcounts Handbook).
Ans.3498


Solution:

12 = 8 + 4 = 7 + 5 = 6 + 6  (no number can be bigger than 8  )

The answer is C(12, 4) *2 + C(12, 5)*2 + C(12,6).

note we multiply C(12,4) and C(12,5) by 2 because they can switch ( for example Mr. Mirus’s class has 8 students and and Ms. Batty’ class has 4 students or Mr. Mirus’s class has 4 students and and Ms. Batty’ class has 8 students)

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