There are $8$ guitars on display, of which $3$ are... | Free GMAT Practice Question | GMAT Panda

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Question

There are 8 guitars on display, of which 3 are electric and 5 are acoustic. How many possible selections of 4 guitars contain at least one electric and at least one acoustic?

Answer Choices

A. $55$
B. $60$
C. $65$
D. $70$
E. $75$

Steps

Explanation	Calculations	Help
$We want selections containing "at least one electric and at least one acoustic", so we count all possible ways to choose 4 guitars and later subtract the cases with all guitars of a single type.$
$We apply the "Combination Formula" to find the total number of ways to choose 4 guitars from 8.$	$C (8, 4) = \frac{8 !}{4 ! \times ( 8 - 4 )!}$	Theory & Tactics Method Card PC3-C Click to view full details
$We simplify the factorial ratio by canceling common terms.$	$= \frac{8 \times 7 \times 6 \times 5 \times 4 !}{4 ! \times 4 \times 3 \times 2 \times 1}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We continue to cancel common factors and compute the result.$	$= \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{2 \times 7 \times 2 \times 5}{2 \times 1} = \frac{2 \times 7 \times 2 \times 5}{2 \times 1} = 7 \times 2 \times 5 = 70$	Theory & Tactics Method Card CALC2-C Click to view full details
$We apply the "Combination Formula" to count the number of ways to choose 4 acoustic guitars from the 5 available.$	$C (5, 4) = \frac{5 !}{4 ! \times 1 !} = \frac{5 \times 4 !}{4 ! \times 1 !} = 5$	Theory & Tactics Method Card PC3-C Click to view full details
$We apply the "Combination Formula" to count the number of ways to choose 4 electric guitars from the 3 available; since 4 is more than 3, there are no such selections.$	$C (3, 4) = 0$	Theory & Tactics Method Card PC3-C Click to view full details
$We apply the "Combinatorics Inverse Rule" to subtract the all-electric and all-acoustic cases from the total.$	$70 - (0 + 5) = 65$	Theory & Tactics Method Card PC2-C Click to view full details

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Final Answer

C

Question

There are 8 guitars on display, of which 3 are electric and 5 are acoustic. How many possible selections of 4 guitars contain at least one electric and at least one acoustic?

Answer Choices

A. $55$
B. $60$
C. $65$
D. $70$
E. $75$

Steps

Explanation	Calculations	Help
$We want selections containing "at least one electric and at least one acoustic", so we count all possible ways to choose 4 guitars and later subtract the cases with all guitars of a single type.$
$We apply the "Combination Formula" to find the total number of ways to choose 4 guitars from 8.$	$C (8, 4) = \frac{8 !}{4 ! \times ( 8 - 4 )!}$	Theory & Tactics Method Card PC3-C Click to view full details
$We simplify the factorial ratio by canceling common terms.$	$= \frac{8 \times 7 \times 6 \times 5 \times 4 !}{4 ! \times 4 \times 3 \times 2 \times 1}$	Theory & Tactics Method Card CALC2-C Click to view full details
$We continue to cancel common factors and compute the result.$	$= \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{2 \times 7 \times 2 \times 5}{2 \times 1} = \frac{2 \times 7 \times 2 \times 5}{2 \times 1} = 7 \times 2 \times 5 = 70$	Theory & Tactics Method Card CALC2-C Click to view full details
$We apply the "Combination Formula" to count the number of ways to choose 4 acoustic guitars from the 5 available.$	$C (5, 4) = \frac{5 !}{4 ! \times 1 !} = \frac{5 \times 4 !}{4 ! \times 1 !} = 5$	Theory & Tactics Method Card PC3-C Click to view full details
$We apply the "Combination Formula" to count the number of ways to choose 4 electric guitars from the 3 available; since 4 is more than 3, there are no such selections.$	$C (3, 4) = 0$	Theory & Tactics Method Card PC3-C Click to view full details
$We apply the "Combinatorics Inverse Rule" to subtract the all-electric and all-acoustic cases from the total.$	$70 - (0 + 5) = 65$	Theory & Tactics Method Card PC2-C Click to view full details

Scroll horizontally to view all columns

Final Answer

C