A football team has 6 tall defenders and 5 short defenders. How many ways can a group of defenders be chosen if at least one tall defender must be included?

Question

Asked on May 14, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-05-14T19:41:36.324Z

Hey Hurry, good to see you again. Here's how to solve question 11(b): Let N_T be the number of tall defenders and N_S be the number of short defenders. We have N_T = 6 tall defenders and N_S = 5 short defenders. The problem asks for the number of ways to choose a group of defenders based on certain conditions. Since the total number of defenders to be chosen is not specified, we consider all possible non-empty groups that satisfy the conditions. i) At least one tall defender Step 1: Calculate the number of ways to choose at least one tall defender from 6 tall defenders. The total number of ways to choose any number of tall defenders from 6 is 2^6. This includes the case of choosing zero tall defenders. To choose at least one tall defender, we subtract the case where zero tall defenders are chosen: Ways for tall defenders = 2^N_T - N_T0 = 2^6 - 1 = 64 - 1 = 63 Step 2: Calculate the number of ways to choose any number of short defenders from 5 short defenders. Since there are no restrictions on short defenders for this part, we can choose any number from 0 to 5: Ways for short defenders = 2^N_S = 2^5 = 32 Step 3: Multiply the ways for tall and short defenders to get the total number of ways. Total ways = (Ways for tall defenders) × (Ways for short defenders) = 63 × 32 = 2016 The number of ways to choose at least one tall defender is 2016. ii) At most two short defenders Step 1: Calculate the number of ways to choose at most two short defenders from 5 short defenders. This means choosing 0, 1, or 2 short defenders: Ways for short defenders = 50 + 51 + 52 = 1 + 5 + (5 × 4)/(2 × 1) = 1 + 5 + 10 = 16 Step 2: Calculate the number of ways to choose any number of tall defenders from 6 tall defenders. Since there are no restrictions on tall defenders for this part, we can choose any number from 0 to 6: Ways for tall defenders = 2^N_T = 2^6 = 64 Step 3: Multiply the ways for short and tall defenders to get the total number of ways. Total ways = (Ways for short defenders) × (Ways for tall defenders) = 16 × 64 = 1024 The number of ways to choose at most two short defenders is 1024. Send me the next one 📸

Accepted Answer · 2026-05-14T19:41:36.324Z

Hey Hurry, good to see you again. Here's how to solve question 11(b): Let N_T be the number of tall defenders and N_S be the number of short defenders. We have N_T = 6 tall defenders and N_S = 5 short defenders. The problem asks for the number of ways to choose a group of defenders based on certain conditions. Since the total number of defenders to be chosen is not specified, we consider all possible non-empty groups that satisfy the conditions. i) At least one tall defender Step 1: Calculate the number of ways to choose at least one tall defender from 6 tall defenders. The total number of ways to choose any number of tall defenders from 6 is 2^6. This includes the case of choosing zero tall defenders. To choose at least one tall defender, we subtract the case where zero tall defenders are chosen: Ways for tall defenders = 2^N_T - N_T0 = 2^6 - 1 = 64 - 1 = 63 Step 2: Calculate the number of ways to choose any number of short defenders from 5 short defenders. Since there are no restrictions on short defenders for this part, we can choose any number from 0 to 5: Ways for short defenders = 2^N_S = 2^5 = 32 Step 3: Multiply the ways for tall and short defenders to get the total number of ways. Total ways = (Ways for tall defenders) × (Ways for short defenders) = 63 × 32 = 2016 The number of ways to choose at least one tall defender is 2016. ii) At most two short defenders Step 1: Calculate the number of ways to choose at most two short defenders from 5 short defenders. This means choosing 0, 1, or 2 short defenders: Ways for short defenders = 50 + 51 + 52 = 1 + 5 + (5 × 4)/(2 × 1) = 1 + 5 + 10 = 16 Step 2: Calculate the number of ways to choose any number of tall defenders from 6 tall defenders. Since there are no restrictions on tall defenders for this part, we can choose any number from 0 to 6: Ways for tall defenders = 2^N_T = 2^6 = 64 Step 3: Multiply the ways for short and tall defenders to get the total number of ways. Total ways = (Ways for short defenders) × (Ways for tall defenders) = 16 × 64 = 1024 The number of ways to choose at most two short defenders is 1024. Send me the next one 📸

A football team has 6 tall defenders and 5 short defenders. How many ways can a group of defenders be chosen if at least one tall defender must be included?

Need help with your own homework?

More Mathematics Questions

A football team has 6 tall defenders and 5 short defenders. How many ways can a group of defenders be chosen if at least one tall defender must be included?

Need help with your own homework?

More Mathematics Questions