A certain lock has three tumblers, and each tumbler can assume seven positions. How many different possibilities are there?

Try 7 times for the first, 7 times for the second 7 times for the third, the possibilities (3 independent trials each of 7 outcomes) is 7³=343.

A certain lock has three tumblers, and each tumbler can assume six positions. How many different possibilities are there?

To determine the number of different possibilities for a lock with three tumblers, where each tumbler can assume seven positions, we can use the concept of permutations.

A permutation is an arrangement of objects in a specific order. In this case, each position of the tumbler represents an object in the arrangement. Since there are seven positions for each tumbler, there are seven objects to arrange in each case.

To find the total number of possibilities, we need to find the number of permutations for each tumbler and multiply them together.

Since each tumbler can assume seven positions, the number of permutations for each tumbler is 7.

To calculate the total number of possibilities, we multiply the number of permutations for each tumbler together:

Number of possibilities = Number of permutations for tumbler 1 * Number of permutations for tumbler 2 * Number of permutations for tumbler 3

Number of possibilities = 7 * 7 * 7 = 343

Therefore, there are 343 different possibilities for the given lock with three tumblers, where each tumbler can be in one of seven positions.