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Question

Difficulty: EasyCombinations

A fruit vendor has 77 distinct types of fresh fruits on display. A customer wants to buy a gift basket containing exactly 44 different types of fruits. How many different combinations of fruits can the customer choose?

Answer: 35 ways

Answer

The customer can choose the fruits in 35 different ways.
The number of ways to choose r=4r = 4 items from n=7n = 7 distinct items without regard to order is given by 7C4=7!4!3!=2106=35^7C_4 = \frac{7!}{4!3!} = \frac{210}{6} = 35.

Step-by-Step Solution

1
Identify total elements (nn) and selected subset size (rr).
n=7n = 7 and r=4r = 4.
Since the selection order inside the fruit basket does not matter, combinations (nCrnCr) must be used.
2
Substitute values into the combination formula nCr=n!r!(n−r)!^nC_r = \frac{n!}{r!(n-r)!}.
7C4=7!4!(7−4)!=7!4!3!^7C_4 = \frac{7!}{4!(7-4)!} = \frac{7!}{4!3!}.
This evaluates the total ways to choose 4 items from 7 without repetition or ordering.
3
Simplify the factorials and compute the numerical result.
7×6×53×2×1=35\frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35.
Canceling out 4!4! from numerator and denominator gives (7×6×5)/6=35(7 \times 6 \times 5) / 6 = 35.

Key Concept

Combinations (nCrnCr)
Estimated Time:45s
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Topics

Combinations

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