Question

Given below are two statements :
Statement I: The number of ways to pack six copies of the same book into four identical boxes where a box can contain as many as six books, is 9.
Statement II: The minimum number of students needed in a class to guarantee that there are at least six students whose birthday fall in the same month, is 61 . In the light of the above statements, choose the correct answer from the options given below

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is A

To determine whether both statements given are true, let's analyze each separately:

1. Statement I: The number of ways to pack six copies of the same book into four identical boxes where a box can contain as many as six books, is 9.
   • This is a problem of distributing identical items (books) into identical bins (boxes). This can be visualized as a partition problem where we need to partition the number 6 with at most 4 parts.
   • Using the partition theory, we calculate the number of partitions of 6 with up to 4 parts. The partitions are:
     • 6,
     • 5 + 1,
     • 4 + 2,
     • 4 + 1 + 1,
     • 3 + 3,
     • 3 + 2 + 1,
     • 3 + 1 + 1 + 1,
     • 2 + 2 + 2,
     • 2 + 2 + 1 + 1.
   • Counting these, we get 9 partitions.

   Hence, Statement I is true.

2. Statement II: The minimum number of students needed in a class to guarantee that there are at least six students whose birthday falls in the same month, is 61.
   • There are 12 months in a year. To ensure that at least 6 students share the same birth month, we use the pigeonhole principle.
   • According to the pigeonhole principle, if there are \( n \) students and 12 months, the minimal configuration that avoids any month having 6 students would distribute 5 students in each of the 12 months, totaling \( 5 \times 12 = 60 \) students.
   • To guarantee at least 6 students in one month, the 61st student will push some month beyond 5 students. Thus, the minimum number required is 61.

   Hence, Statement II is true.

Overall conclusion: Both Statement I and Statement II are true.