Combinatorial Analysis of Marble Selection

Given a jar containing 7 green marbles and 9 blue marbles, we need to determine the number of different groups of 4 marbles that can be selected. This problem can be solved using combinatorics, a branch of mathematics that deals with counting and arranging elements in sets. The solution involves calculating various combinations based on how many green and blue marbles are selected.

Combinations of Marbles

The problem can be broken down into several cases based on the number of green marbles selected:

4 Green and 0 Blue

The number of ways to choose 4 green marbles from 7 is denoted as C(7, 4). Using the combination formula, we get:

C(7, 4) frac{7!}{4!(7-4)!} frac{7times6times5}{3times2times1} 35

3 Green and 1 Blue

The number of ways to choose 3 green marbles from 7 and 1 blue marble from 9 is denoted as C(7, 3) times C(9, 1). Calculating this, we have:

C(7, 3) frac{7!}{3!(7-3)!} 35

C(9, 1) frac{9!}{1!(9-1)!} 9

Therefore, the total number of ways is 35 times 9 315.

2 Green and 2 Blue

The number of ways to choose 2 green marbles from 7 and 2 blue marbles from 9 is denoted as C(7, 2) times C(9, 2). Calculating this, we have:

C(7, 2) frac{7!}{2!(7-2)!} 21

C(9, 2) frac{9!}{2!(9-2)!} 36

Therefore, the total number of ways is 21 times 36 756.

1 Green and 3 Blue

The number of ways to choose 1 green marble from 7 and 3 blue marbles from 9 is denoted as C(7, 1) times C(9, 3). Calculating this, we have:

C(7, 1) frac{7!}{1!(7-1)!} 7

C(9, 3) frac{9!}{3!(9-3)!} 84

Therefore, the total number of ways is 7 times 84 294.

0 Green and 4 Blue

The number of ways to choose 4 blue marbles from 9 is denoted as C(9, 4). Calculating this, we have:

C(9, 4) frac{9!}{4!(9-4)!} 126

Total Combinations

Adding all the combinations together, the total number of different groups of 4 marbles is:

35 315 756 294 126 1491

Order Consideration

Depending on whether the order of the marbles is important or not, the number of possibilities differs. If the order of selection does not matter, then there are 5 possibilities based on the number of green and blue marbles selected. If the order is important, the number of possibilities is 16.

No Order

GBBB GGBB GGGB GGGG BBBB

Order Matters

GGGG BGGG, GBGG, GGBG, GGGB GGBB, GBGB, GBBG, BBGG, BGBG, BGGB GBBB, BGBB, BBGB, BBBG BBBB

This detailed analysis and its mathematical calculations show the extensive use of combinatorial methods in solving real-world problems such as selecting marbles from a jar. The problem-solving techniques and the use of combination formulas provide a clear understanding of the underlying principles.