If a mother has three bananas, two pears, and two oranges, in how many different ways can she give the fruit to her daughter in one week, one piece of fruit per day? Be sure to say how you could use this with your students and mention any adaptations that may be necessary.

Question

If a mother has three bananas, two pears, and two oranges, in how many different ways can she give the fruit to her daughter in one week, one piece of fruit per day? Be sure to say how you could use this with your students and mention any adaptations that may be necessary.

Answer 1

number of ways = 7!/(3!2!2!) = 210

Answer 2

First I'd say we go ahead and make the fruits into letters:

b = bananas
p = pears
o = oranges

Answer 3

To determine the number of different ways the mother can give the fruit to her daughter, we need to consider the number of ways she can give each type of fruit on each day, and then find the product of those possibilities.

Let's break it down:

1) Bananas: The mother has 3 bananas and 7 days in a week. On any given day, the mother can choose one banana out of the three available. So the number of ways she can give the bananas to her daughter throughout the week is 3 * 3 * 3 * 3 * 3 * 3 * 3 = 3^7.

2) Pears: Similarly, the mother has 2 pears and 7 days in a week. On any given day, she can choose one pear out of the two available. So the number of ways she can give the pears to her daughter throughout the week is 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^7.

3) Oranges: The mother has 2 oranges and 7 days in a week. On any given day, she can choose one orange out of the two available. So the number of ways she can give the oranges to her daughter throughout the week is 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^7.

To find the total number of ways, we multiply the three results together as each fruit choice is independent:

Total ways = 3^7 * 2^7 * 2^7 = 3^7 * (2^7)^2 = 2187 * 128 * 128 = 44,06,336.

So, the mother can give the fruit to her daughter in 44,06,336 different ways throughout the week.

With students, you can explain this problem by breaking it down into smaller sub-problems, one for each type of fruit, and then combining the results. Furthermore, you can provide variations to this problem by changing the number of fruits or the number of days, which would require adjusting the calculations accordingly. This exercise can help students understand the concept of permutations and combinations, as well as strengthen their skills in multiplication and exponentiation.