A quality team computes combinations of samples for an inspection lot. Given total items (n) = 7.0000; items chosen (r) = 2.0000, determine the number of combinations (C).

Start with the thinking

• The governing relation printed in this handbook section is Combinations.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Combinations count the unordered subsets of r items chosen from n distinct items.

Step-by-step solution

1. Step 1 — State the governing relation:

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

2. Step 2 — Rearrange symbolically for C:

   $$C = \dfrac{n!}{r!(n-r)!}$$

3. Step 3

   $$List the givens: total items (n) = 7.0000, items chosen (r) = 2.0000$$

4. Step 4 — Substitute the given values:

   $$C = \dfrac{7.0000!}{2.0000!(7.0000-2.0000)!}$$

5. Step 5 — Evaluate:

   $$C = 21.0000$$

6. Step 6 — Check: returning C = 21.0000 to

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 42.0000 — kept a factor of two that cancels in the correct rearrangement.
• 10.5000 — dropped that same factor in the other direction.
• 23.1000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Permutations and Combinations

A statistician finds the standard deviation of a sum of independent random variables. Given std dev of X (sX) = 7.3000; std dev of Y (sY) = 1.5000, determine the std dev of Z = X+Y (sZ).

Start with the thinking

• The governing relation printed in this handbook section is Combinations of random variables.
• Everything except sZ is given, so isolate sZ symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• For independent combinations of random variables, the variance of their sum equals the sum of variances.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

2. Step 2 — Rearrange symbolically for sZ:

   $$sZ = \sqrt{\sigma_X^2+\sigma_Y^2}$$

3. Step 3

   $$List the givens: std dev of X (sX) = 7.3000, std dev of Y (sY) = 1.5000$$

4. Step 4 — Substitute the given values:

   $$sZ = \sqrt{\sigma_X^2+\sigma_Y^2}$$

5. Step 5 — Evaluate:

   $$sZ = 7.4525$$

6. Step 6 — Check: returning sZ = 7.4525 to

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 14.9050 — kept a factor of two that cancels in the correct rearrangement.
• 3.7263 — dropped that same factor in the other direction.
• 8.1978 — rounded an intermediate value before the final step.

Reference: FE Handbook — Combinations of Random Variables

A student determines the combinations of committee members from a pool. Given total items (n) = 10.0000; items chosen (r) = 3.0000, determine the number of combinations (C).

Start with the thinking

• The governing relation printed in this handbook section is Combinations.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Combinations count the unordered subsets of r items chosen from n distinct items.

Step-by-step solution

1. Step 1 — State the governing relation:

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

2. Step 2 — Rearrange symbolically for C:

   $$C = \dfrac{n!}{r!(n-r)!}$$

3. Step 3

   $$List the givens: total items (n) = 10.0000, items chosen (r) = 3.0000$$

4. Step 4 — Substitute the given values:

   $$C = \dfrac{10.0000!}{3.0000!(10.0000-3.0000)!}$$

5. Step 5 — Evaluate:

   $$C = 120.0$$

6. Step 6 — Check: returning C = 120.0 to

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 240.0 — kept a factor of two that cancels in the correct rearrangement.
• 60.0000 — dropped that same factor in the other direction.
• 132.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Permutations and Combinations

The combination of random variables from two production lines is analyzed. Given std dev of Y (sY) = 8.5000; std dev of Z = X+Y (sZ) = 13.5200, determine the std dev of X (sX).

Start with the thinking

• The governing relation printed in this handbook section is Combinations of random variables.
• Everything except sX is given, so isolate sX symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• For independent combinations of random variables, the variance of their sum equals the sum of variances.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

2. Step 2 — Rearrange symbolically for sX:

   $$sX = \sqrt{\sigma_Z^2-\sigma_Y^2}$$

3. Step 3

   $$List the givens: std dev of Y (sY) = 8.5000, std dev of Z = X+Y (sZ) = 13.5200$$

4. Step 4 — Substitute the given values:

   $$sX = \sqrt{\sigma_Z^2-\sigma_Y^2}$$

5. Step 5 — Evaluate:

   $$sX = 10.5138$$

6. Step 6 — Check: returning sX = 10.5138 to

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 21.0276 — kept a factor of two that cancels in the correct rearrangement.
• 5.2569 — dropped that same factor in the other direction.
• 11.5652 — rounded an intermediate value before the final step.

Reference: FE Handbook — Combinations of Random Variables

The number of combinations of defective units in a batch is evaluated. Given total items (n) = 6.0000; items chosen (r) = 3.0000, determine the number of combinations (C).

Start with the thinking

• The governing relation printed in this handbook section is Combinations.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Combinations count the unordered subsets of r items chosen from n distinct items.

Step-by-step solution

1. Step 1 — State the governing relation:

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

2. Step 2 — Rearrange symbolically for C:

   $$C = \dfrac{n!}{r!(n-r)!}$$

3. Step 3

   $$List the givens: total items (n) = 6.0000, items chosen (r) = 3.0000$$

4. Step 4 — Substitute the given values:

   $$C = \dfrac{6.0000!}{3.0000!(6.0000-3.0000)!}$$

5. Step 5 — Evaluate:

   $$C = 20.0000$$

6. Step 6 — Check: returning C = 20.0000 to

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 40.0000 — kept a factor of two that cancels in the correct rearrangement.
• 10.0000 — dropped that same factor in the other direction.
• 22.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Permutations and Combinations

An engineer combines two independent random variables representing tolerance stack-up. Given std dev of X (sX) = 9.0000; std dev of Z = X+Y (sZ) = 2.5100, determine the std dev of Y (sY).

Start with the thinking

• The governing relation printed in this handbook section is Combinations of random variables.
• Everything except sY is given, so isolate sY symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• For independent combinations of random variables, the variance of their sum equals the sum of variances.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

2. Step 2 — Rearrange symbolically for sY:

   $$sY = \sqrt{\sigma_Z^2-\sigma_X^2}$$

3. Step 3

   $$List the givens: std dev of X (sX) = 9.0000, std dev of Z = X+Y (sZ) = 2.5100$$

4. Step 4 — Substitute the given values:

   $$sY = \sqrt{\sigma_Z^2-\sigma_X^2}$$

5. Step 5 — Evaluate:

   $$sY = 0.0100$$

6. Step 6 — Check: returning sY = 0.0100 to

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0200 — kept a factor of two that cancels in the correct rearrangement.
• 0.0050 — dropped that same factor in the other direction.
• 0.0110 — rounded an intermediate value before the final step.

Reference: FE Handbook — Combinations of Random Variables

A quality team computes combinations of samples for an inspection lot. Given total items (n) = 12.0000; items chosen (r) = 3.0000, determine the number of combinations (C).

Start with the thinking

• The governing relation printed in this handbook section is Combinations.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Combinations count the unordered subsets of r items chosen from n distinct items.

Step-by-step solution

1. Step 1 — State the governing relation:

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

2. Step 2 — Rearrange symbolically for C:

   $$C = \dfrac{n!}{r!(n-r)!}$$

3. Step 3

   $$List the givens: total items (n) = 12.0000, items chosen (r) = 3.0000$$

4. Step 4 — Substitute the given values:

   $$C = \dfrac{12.0000!}{3.0000!(12.0000-3.0000)!}$$

5. Step 5 — Evaluate:

   $$C = 220.0$$

6. Step 6 — Check: returning C = 220.0 to

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 440.0 — kept a factor of two that cancels in the correct rearrangement.
• 110.0 — dropped that same factor in the other direction.
• 242.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Permutations and Combinations

A statistician finds the standard deviation of a sum of independent random variables. Given std dev of X (sX) = 1.3000; std dev of Y (sY) = 3.6000, determine the std dev of Z = X+Y (sZ).

Start with the thinking

• The governing relation printed in this handbook section is Combinations of random variables.
• Everything except sZ is given, so isolate sZ symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• For independent combinations of random variables, the variance of their sum equals the sum of variances.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

2. Step 2 — Rearrange symbolically for sZ:

   $$sZ = \sqrt{\sigma_X^2+\sigma_Y^2}$$

3. Step 3

   $$List the givens: std dev of X (sX) = 1.3000, std dev of Y (sY) = 3.6000$$

4. Step 4 — Substitute the given values:

   $$sZ = \sqrt{\sigma_X^2+\sigma_Y^2}$$

5. Step 5 — Evaluate:

   $$sZ = 3.8275$$

6. Step 6 — Check: returning sZ = 3.8275 to

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 7.6551 — kept a factor of two that cancels in the correct rearrangement.
• 1.9138 — dropped that same factor in the other direction.
• 4.2103 — rounded an intermediate value before the final step.

Reference: FE Handbook — Combinations of Random Variables

A student determines the combinations of committee members from a pool. Given total items (n) = 7.0000; items chosen (r) = 4.0000, determine the number of combinations (C).

Start with the thinking

• The governing relation printed in this handbook section is Combinations.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Combinations count the unordered subsets of r items chosen from n distinct items.

Step-by-step solution

1. Step 1 — State the governing relation:

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

2. Step 2 — Rearrange symbolically for C:

   $$C = \dfrac{n!}{r!(n-r)!}$$

3. Step 3

   $$List the givens: total items (n) = 7.0000, items chosen (r) = 4.0000$$

4. Step 4 — Substitute the given values:

   $$C = \dfrac{7.0000!}{4.0000!(7.0000-4.0000)!}$$

5. Step 5 — Evaluate:

   $$C = 35.0000$$

6. Step 6 — Check: returning C = 35.0000 to

   $${}_nC_r = \dfrac{n!}{r!(n-r)!}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 70.0000 — kept a factor of two that cancels in the correct rearrangement.
• 17.5000 — dropped that same factor in the other direction.
• 38.5000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Permutations and Combinations

The combination of random variables from two production lines is analyzed. Given std dev of Y (sY) = 2.4000; std dev of Z = X+Y (sZ) = 8.2100, determine the std dev of X (sX).

Start with the thinking

• The governing relation printed in this handbook section is Combinations of random variables.
• Everything except sX is given, so isolate sX symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• For independent combinations of random variables, the variance of their sum equals the sum of variances.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

2. Step 2 — Rearrange symbolically for sX:

   $$sX = \sqrt{\sigma_Z^2-\sigma_Y^2}$$

3. Step 3

   $$List the givens: std dev of Y (sY) = 2.4000, std dev of Z = X+Y (sZ) = 8.2100$$

4. Step 4 — Substitute the given values:

   $$sX = \sqrt{\sigma_Z^2-\sigma_Y^2}$$

5. Step 5 — Evaluate:

   $$sX = 7.8514$$

6. Step 6 — Check: returning sX = 7.8514 to

   $$\sigma_Z^2 = \sigma_X^2 + \sigma_Y^2$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 15.7028 — kept a factor of two that cancels in the correct rearrangement.
• 3.9257 — dropped that same factor in the other direction.
• 8.6365 — rounded an intermediate value before the final step.

Reference: FE Handbook — Combinations of Random Variables